More complete “Play Integrity” would need to come from microG, which would then be available to iodé as well. Currently only “Basic Integrity” is possible. But note that Google changes even make confirming this quite challenging. That post is from one of the core microG devs.
Even without full “Device Integrity”, there are many apps that report they will not work without full “Google Play Services” that do actually work. One of the better places to check is to use Plexus from F-Droid. For example, Mobiles Bezahlen has 2 “Gold” (full functionality) ratings back in August 2025 for microG users of version 7.0.2. But with recent changes to the app it may not be working now as you see.
It really is a bit challenging to reverse-engineer as I suspect you can appreciate. But do take solace that iodéOS has “all the compatiblity” available from microG (and also “a bit more” thanks to signature spoofing and some other parts from ih8sn.