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.
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.