You could try to reflash the rom but with the modem.img from the previous release, no guarantees it will work or won't break anything but might help without a completely clean install
The question is if you could get a push notification once the fp no longer works and the app update is available, not upon opening the updater app manually.
It's not, the app uses the same fp for everyone and so does the basic PIF module.
Only the custom module uses your own harvested fp.
But there's no guarantee it works longer than the standard option.
You can't pass strong with unlocked bootloader, also not needed for gpay.
It seems like Google has banned the fingerprint again, wait for a app update with gpay fix
Maybe you shouldn't use a personal phone for work.
You can't spoof the device and expect your own device name to come up.
Do note that without spoofing services such as Google pay will no longer work.
It was resolved in MIUI 14 builds as well though, several al F5 Pro users reported the standard modem now works and flashing the global one is no longer needed.
You need to add all Playstore and Google services to the denylist.
But both are doing the same thing, if you're rooted and PIF does work might as well use that.