Ask Them, Not the Internet

There is an easy way to avoid doing this. Talk directly to the groups you're concerned about insulting. Explain what you're doing, and explain that you wanted to make sure you didn't accidentally offend them. For example, the Navajo Nation's official homepage would be a good place to start looking for a point of contact.

As long as you are polite, you are very unlikely to offend people simply by asking "is this offensive". (Just don't give off the impression that you don't really care about the answer and just want to hear them say 'no'.)

Have an Author's Note

At the end of the book have a page or so like:

The Sakha are based on several real-life cultures. First and foremost the Puebloans, from whom I copied.... The custom of [X] I got from the [other] people, and [Y] is completely my own invention. [Z] is an actual custom, described in [scholarly source] that I couldn't resist including.

Please don't assume the Puebloans were (in any era) just like the Sakha. Plenty of things had to change for artistic purposes. If you want to know about the real Publoans, I recommend [list of nonfiction books]

See the afterward to 1632 for an example, though that's a little different and has some other stuff too.

Or, if it's a much more light-hearted book, follow Terry Pratchett's example from The Last Continent:

This is not a book about Australia. No, it's about somewhere entirely different which happens to be, here and there, a bit…Australian. Still…no worries, right?

Research, and do your best.

If you can't, as Dayton suggested, actually contact the culture, do as much research as you can. People tend to be inherently understanding, and even though honest mistakes happen, as long as the majority of your work is 'correct' (so to speak) you'll avoid offending the majority of people. In the end, imagine yourself on the other side. If someone decided to write a story with heavy parallels to your culture, but messed up a few side details, would you be offended?

There is only so much you can do if you want to extract many specific, "stereotypical" features from real cultures as set pieces in your work. Using less "real" material and knowing more about the source of your inspiration will be helpful.

That's not the most charitable way of phrasing what you're looking to do, but it's correct in the essentials. There are whole cultures, thousands of years old and rich in detail and nuance beyond routine expression, which have produced the features that have captured your interest.

Extracting a handful of conspicuous features from those cultures without any context and without providing any cultural representation beyond those conspicuous features, just because you think they look cool or suit your setting, generally is what people complain about in situations like this. It's definitely worse if the chosen features denigrate the group, but many object to their cultures being "cheapened" just to suit aesthetics in a fictional setting by someone not at all engaged with their actual culture and history.

So my main answer to the main question is: don't wholesale-copy much from existing cultures, and know enough details to be "inspired" well. Inspired by is not the same as copied from. There are many things that aren't especially culture-specific-- mud houses have been used in many parts of the world at many points in history by totally disparate cultural groups. Mud houses, alone are probably fine.

Mud houses that look exactly like those used in a certain culture... that's a bit different. Architectural styles and practices are varied, and can be very culture-specific. Having mud houses is one thing, but having mud houses that look just like Puebloan mud houses is a bit closer to copying rather than being inspired. This becomes more severe as you add more features which are specific to a given culture.

And just as importantly, think about the features that you aren't carrying over from your inspiration. A big risk is not noticing distinctions meaningful to members of the source culture: "these groups are all the same" is very offensive to groups that view themselves as being very different, especially if you casually blend details together in a way which more familiarity with the cultures would prevent.

If you're using the architecture, clothing, and art styles from a culture but nothing else, it's fairly easy for someone to say that you are caricaturing the culture without bothering to know (or at least, express) anything meaningful about it. This is especially the case if any of those elements had broader cultural significance (like clothing styles that weren't just popular for some reason, but had deeper religious or cultural meaning)-- it may not be possible to respectfully imitate one without the other.

As an example, I have some Italian heritage. I don't especially care about Italian representation in media in the modern day (we don't see the same kinds of social attitudes that were common in the U.S. around the turn of the 20th century), but a cartoon of a mustachioed chef eating spaghetti and meatballs while speaking in a funny accent is a clearly "Italian" caricature, but is a poor representation of the culture that produced the Roman Empire, lead the Renaissance, and countless other items.

As for the "know enough details" piece, a person can't be much inspired by a culture they know little about. Doing some real research on a culture (reading published papers, well-regarded history books, etc.) can give you enough understanding that you aren't just copying superficial details. A few hours of casual googling simply isn't enough.

Broader notes:

The goal of not offending a "culture" is good, but fuzzy. Cultures often don't have representatives empowered to make blanket judgements on this sort of topic, as noted in other answers. And even if they do, that's no defense against individuals feeling offended and complaining. There is no bright line for the amount of respectfulness you can display (however that's defined) which guarantees that no one will feel offended, or that relieves you of any possible responsibility if someone takes offense.

There is no obvious standard for how to do this sort of thing "right", aside from not doing it at all. Talking to people who are a part of the source culture, especially if they are cultural experts (the exact definition of which will vary between groups) to get guidance on how appropriate your ideas seem to them can be helpful, but will never be definitive.

The shallower your inspiration and translation of cultural elements are, the more likely you are to give offense. The more knowledgeable, cautious, and substantive you are in representing cultural elements, the less likely. And if you can avoid copying enough superficial elements to obviously point to a real culture, you'll be on firmer ground still.

Make them different.

https://www.dailymotion.com/video/x72mkoi

I was so impressed with Valley of the Dinosaurs, then and now. The cavemen would have been caricatures of movie amerinds - speaking in clipped "ug" English, dressing like movie amerinds; the whole deal. But: they were blond, with weird mustard complexions, these cavemen, and so not Indians.

Do the same. You are already bringing in Monster Hunter. The idea of adding tipis and totem poles is not outrageous; if you are digging the amerind thing as a basis for a story make a big mix of all the things you think are cool. A stone pueblo totem pool would be very cool. People do that for Western Europe based fantasy and African based fantasy all the time. Mix Pueblo III with elements from older pueblo (Anasazi) periods, other Amerinds elementals you like, and add some Middle Eastern desert tech. No-one will be able to claim you are depicting their people in your fiction because you will not be.

You're worried about offending real world cultures (or, more correctly, people belonging to those cultures). It's worth pointing out that most cases of cultural offense are rooted in the perception that the offender has got something wrong in terms of that culture. They're mis-attributing or misunderstanding something. Something has been used/borrowed/stolen without any tie to it's significance from that originating culture.

This is an easy trap to fall in to when designing a fictional culture by way of picking and choosing from real world cultures. But, this issue can be avoided if you take a different approach to designing the features you want your culture to have. Instead of reverse-engineering them by basing them on real world cultures, I would suggest a different approach.

You are building a world, and you're designing a culture that lives in that world. Instead of basing that culture on real-world cultures, make your culture fit your world. The cultural aspects you seem to be focused on (shelter, food gathering, etc) are all environmentally influenced anyways, so this should be straightforward. If you write your culture such that it fits their world then there's essentially nothing for anyone to be offended by.

In other words, drop the approach of basing your culture on real cultures in our world, and instead base your culture on the world in which they live.

If there's a real world culture that uses adobe brick, it's because their environment and tech supported it. Instead of worrying about whether or not your culture's use of adobe brick will offend a real world culture who used adobe brick, just ensure you are writing your world and your culture such that adobe brick makes sense for your culture. Such consistency is hard to get offended by. And although such an approach can seem overwhelming, since you're taking on the burden of consistency, in the end it's a more organic way to create a believable culture - potential offense aside, the result will likely make more sense to readers in the end than a culture that was just mashed together without any rhyme or reason.

Would another option be (perhaps not for this story, but other worldbuilding) to not just hang a lampshade on it, but have that be part of the point?

Like if I had post-apocalyptic humans spend a few centuries in the Midwest or wherever, and now life is better, they're back to having more leisure time and thinking beyond survival into creating a better world. They know of America as the name of their ancestral land, but details are lost to memory. They, like many, are interested in their heritage and some people are into researching and resurrecting some of it.

THEN they find that parts of Walt Disney World's Magic Kingdom survived -- maybe a big pre-flooding dome protected major parts of it, but the utilidors are permanently blocked off. Finally, they can see images of America in its glory: the Hall of Presidents -- Small World -- Haunted Mansion -- Carousel of Progress! (There's no Smithsonian strip of museums to compare things to). So now this is their artifact-driven attempt to understand How America Was. DisnAmericans believed that their Ancestors frolicked happily in houses to watch future generations. I think in Hall of Presidents, that though the presidents speak in order in a roll-call, they're physically grouped non-chronologically, so one may assume a tricorner hat and breeches could be worn along side a 1970s suit and tie, just different styles. More evidence might be the Family that recurs in Carousel. They clearly never traveled because they believed the world was very small.

Part of this imaginary version works because we know the "mainstream US culture" and we know about Disneyworld's simplifications, and we can imagine the effects of the animatronics appearing as a display without much context.

If you do something like that with a culture you do not know very well (yet), make sure that you are similarly making it clear that you know and communicate that, say, the Sacred Sculpture that your characters have passed down for generations is a toy or simple everyday thing that they are misunderstanding and imbuing with unnecessary reverence.

So this is advanced-level worldbuilding, especially when the building blocks come from a culture you don't understand. But finding creative work BY and FOR the groups you're writing about (movies, novels, stories) -- things where they don't care if outsiders get the jokes or references -- that can help you identify potential elements that are more safe to explore. (still research: It may be like "I can pick on my brother, my cousin can pick on my brother, but if an outsider picks on my brother, we all team up!")

Learn from Real Life example: Buddhism

Buddhism was started by "Siddhartha Gautam", who was a Hindu and Buddhism itself can be thought as a subset of Hinduism without all its god level complexities, just its spiritual intensity.

You can start investigating the origins of Buddhism and how he extracted the best parts of Hinduism out of it and created a whole new culture without offending anyone.