Is it a bad idea to let them know I've done this?

It's a terrible idea (unless you asked for and received permission beforehand).

Could I potentially try to sell it or license it to them, or maybe
show it during the interview in case I ever decide to apply for a job
at their company?

You want to sell them their own mobile app?

Talk to your lawyer before even thinking about doing this.

Please keep in mind that this company is in the financial services
industry.

Even more reason to remain silent. Financial Services company tend not to fool around much.

Don't share the app - but if thoughtfully done, you can share ideas for the better interface

Depending on their priorities, companies may or may not focus on all aspects of the tool they are providing. Coming up with a good UI can be challenging in niche industries. The way I see it, you are a customer who felt pained by the existing interface, and rather than just cribbing about it, were able to reverse engineer and build a better solution. However, Your innovation is limited to the better interface, and not to the core backend or APIs that the app is using, or the features that it may be providing.

Is it a bad idea to let them know I've done this? Could I potentially try to sell it or license it to them

So yes, publishing the app / reselling it back / even sharing it among friends / letting them know is a terrible idea like others have pointed out, both for being unethical and illegal.

IANAL, but most apps have a terms of use or about section or legal EULA terms, that will declare the original app a copyrighted, trademarked product of the publisher - so you would end up violating the intellectual property laws of the land on distribution. Similarly, app stores have their own policies for such unauthorized ripoffs. Similarly, I find distributing a product that uses someone else's IP without permission unethical.

maybe show it during the interview in case I ever decide to apply for a job at their company?

However, you can share the wiremocks for the new and improved interface - using a tool such as Invision (or whatever helps you explain workflow better). This is after all a piece of your innovation. This helps let the company know that there are better ways of presenting their services to their customers. Depending on the size of the company and the kind of people involved, it can be a plus for you during interviews as well.

This is actually a common interview question for app developers, UX designers, product managers - how would you improve the existing app/product - so if you are targeting similar role, you will be at an advantage if you present the information thoughtfully.

Will probably be bad, almost certainly will not be good.

If they have the slightest shred of competence as a financial firm, I can tell you now that they will not use your app.

Why? Because it will be more expensive to do security checks and reviews of the entirety of your codebase (something I'll wager you have not done yourself) than to simply reimplement the features in-house.

As for using this in a job interview, you are even less likely to get favourable results. You will be essentially pointing out that you discovered unauthorised software can access their closed services (a very serious security vulnerability) and instead of disclosing it responsibly, you let it sit around for god-knows-how-long, and you used it to your own ends. That is not a good look. In fact you are admitting to be in breach of common hacking laws (CFAA in the USA, CMA in the UK, probably a few others in other countries).

Finally, you will be admitting to breaking their TOS. That gives them grounds to give you the boot as a customer, and depending on what features your app has that isn't on the official app, may even be a very realistic course of action (e.g: if your app does something that goes against the spirit of the service, like the Casper app for Snapchat did)

The only thing that may be worth letting them know about is potential security problems with their API. This is a financial services company, but you were able to reverse engineer the app, presumably authenticate with a service and retrieve details.

This may be a problem (hard to tell as you provide no detail of what it does) and they may thank you for details of a security issue you have found (without necessarily telling them how it was found).

Note - not all companies will thank you for this, and some may actively frown upon it (if the latter, may be an idea to look for a new provider). It's worth looking to see if they have a service to report these items before blindly making contact.

You could just ask the company bosses/directors/managers if they would be happy for you to reverse engineer their app and build a much better version of it, free of charge - mentioning that you have the skills to do that and know for sure that it could be made a lot better. They will almost certainly know already that their mobile app sucks and be interested in talking to you. If they don't respond or aren't happy, just keep quiet and move on to dealing with organisations that are more likely to succeed.

My guess is that they would be a lot happier if you solved their significant problem for them and would then have a high opinion of your value as a coder. So ask if they'd be okay with it, without telling them you have already done it.

No. When a development team within a company develops a product they have certain tools/processes that they use to do it (including testing). If you did not use similar things then incorporating your version of the product into their organization would definitely be more expensive than them not doing anything since now they have to hire/train people to work with your way of doing things. I doubt their tech leaders would want you to essentially be dictating how they operate. There's probably a business driven reason why their app sucks like they didn't think it was worth it to invest more money in making it better and they're working on other aspects of things behind the scenes or focusing on marketing. That thinking isn't going to change because you show up and say, 'Hey pay me money and I'll give you this.'

Most likely outcome is that they would have their lawyer send you an order to stop reverse engineering stuff & to stop using what you created which puts you in a worse position than you're in now.

Your best move is to find out a legal way to have your own app service/business and use your expertise to make money there.