People always bring up online banking, and its so fascinating to me; I've basically always (since 2010 at least) done banking online using mobile web. Are there any institutions that require using an app? For one of my institutions, the app is just a wrapper around the mobile site anyway.
I made the switch to GrapheneOS recently and its been quite pleasant overall, relative to expectations.
Everything else just works, barring notifications for some apps and things that are obviously Google-oriented (e.g. google camera). Also, location is seriously degraded, because it actually _only_ works with GNSS, as far as I can tell.
I've been making the switch to sites pinned to home screen for things that aren't useful offline (banking, weather) and that's been working well, but largely unnecessary. I just really don't trust Android apps generally.
In most of the cases either GrapheneOS + sandboxed Play Services or CalyxOS + MicroG works just fine with banking apps or any other apps.
What's not working: apps using strict SafetyNet checks, in-app purchases (on MicroG), GPlay license checks (on MicroG). And rest assured, location services will be a real pain in your ass on Graphene (and sometimes DejaVu may deadlock on Calyx, better turn it off completely and replace with "GSM Location" Unified NLP backend).
Every bank app is different. I'd advise getting a cheap phone which supports LineageOS (ROM with the widest device support) and see if it works there first. You can also try LineageOS with microG builds to see if it only needs basic verification with Play Services in order to work.