I ran a test yesterday with my PS5 and a Quadstick + FGC2.
I updated the firmware in the FGC2 to the latest version (1.1.3.1) and ran the Quadstick with the FGC2 for a couple of hours on my PS5 and it worked the whole time.
Brook's comment for version 1.1.3.1 is "1. Test PS5 connection stability" so I think they have been experiencing disconnection problems. They greatly trimmed down the number of versions they make available and the one prior, 1.1.2.7 said "For testing purposes only, to ensure stable online connectivity with the PS5 console.(No vibration function)".
We don't need vibration or trigger feedback and can turn those off in the PS5 settings. That will reduce the number of messages being sent from the PS5 to the FGC2, if the FGC2 is having trouble with those.
These are the settings on the PS5 that I used:

Also, set the time until controllers turn off to "don't turn off".
and I set the console to not enter rest mode while playing games:
Bluetooth on the PS5 must be turned off:
The other thing I did was set the volume of the speaker on the controller to 0.