I have the same exact issue also on a P6P.
Mine will say "press firmly on sensor" for a split second before it fails. Subsequent attempts work no problem. Only happens on AOD screen after it sits for a few seconds after being locked (when it goes down to what I think is low refresh rate to save battery).
This started on one of the last Android 13 builds (I switched to Graphene in August and it didn't have this bug then) and persists to the 2023102300 build I'm on.
Have tried turning off security mechanisms such as PIN scramble, auto-reboot, etc (anything I can find that reads from or has anything to do with any kind of lock screen authentication) to no avail.
Seeing others specifically with P6P are having this issue, makes me think it is a bug affecting just this phone.
Outside of this the fingerprint actually works very reliably considering the reputation this reader has. It took me a while to tame it lol but accuracy is actually very good. But this bug is very frustrating and I have just gotten in the habit of either raising the phone first, tapping the screen first, or hitting the lock button to take it out of the deep AOD screen before using the fingerprint.