So could you use the AP via the touchscreen?
My AP intermittently was cutting out earlier in the year similar you what you describe and I ended up with the new servos (just before the program to replace them was announced). Has been fine since. Limited information from Garmin as to whether it was the issue or not, they just looked at the file and said new servos all round…
WRT the heat, I suspect most of the hardware can cope with the typical heat scenarios encountered in service although if the GMC is up on the glareshield I can see that getting very hot in an parked aircraft. I still doubt hot enough to kill it tho.
In my RV7 in summer my old GNS430s bezel would get too hot to touch in summer in flight.
I have noticed once in the peak of summer I got into the plane on a 40+c day after it had been parked all afternoon and the GDU was dim at startup. Never seen it again but I assumed at the time it was some sort of heat related protection. I didn’t fly so can’t confirm if it would have resolved. It was fine the next morning.
I think you have a servo failure.