I’m on an Xperia 10 iii with SFOS 4.4.0.68 and Microg (but no Google Play Services).
Both BancoPosta and Poste Italiane complain about missing google services but work (I don’t use BancoPosta for payments, so I can’t test notifications, but the apps seems to work just fine). With Poste Italiane you can’t search for the nearest post office (because of missing google services).
For some reason, though, PosteID doesn’t work, right after launch it complains that there is no internet connection (but there is) or that it is not allowed to connect (but I was never asked nor denied access to the internet to the app itself).