I already run LineageOS, but with Play services. I would like to be able to ditch Play services, but still need the Play store for things like my banking app, and an app to log in to government services.
I'd say it's a toss up whether specific apps will definitely work. But if they don't I'd recommending segmenting between different physical devices, and making the one that lives in your pocket as secure as possible. It's likely that you don't need to run banking and government apps on the same device that's privy to your movement.
And banking / government apps tend to work in Europe (at least the ones I have tried). Notable exceptions for me are Revolut (shame!) and McDonalds (who knew microG is the healthier option haha). Of course, in the US things might be vastly different.
These mobile payments only work with your banks app or a dedicated app (Payconiq).
My current approach is to put all these apps in my work profile which I can turn off (using Insular from F-Droid). Only apps for which I need background activity or instant notifications (Signal, an open source podcast app, and sadly WhatsApp) are installed in the main profile.
Sadly, this approach still requires me to have Google services always running in the background for a functioning Play store in my work profile.