I’ve had a few z height issues, and fixed a few issues, but I’m not sure they are the same issues you’re having.
My understanding of how this works is when the machine boots with the laser module on it switches from G53 native space into a G54 work offset with the last saved work offset read from EEPROM. Additionally the laser focal length is stored in the toolhead’s memory and read at boot.
If you start a print from the touchscreen is usually (but sometimes doesn’t) reset the work offset in G54 to X/2, Y/2, and Z=laser focal length as saved in the touchscreen.
I’m not really sure what exactly you have tried already. Maybe try adjusting the laser focal length in the touchscreen so that it’s re-written in the toolhead, then going through the touchscreen wizard to start a job from the touchscreen. At the “set work origin” part if it looks right you can back out and power cycle the machine, as that should be saved in the EEPROM.
If it doesn’t look right, sometimes I’ve had luck power cycling the machine and trying again.
I see you’re playing around with G92, and I’ve also tried that and haven’t really had luck with that as a permanent fix as the touchscreen just ignores it and resets the work offset at every job start.