If fridge is set on auto, and you are plugged into shore power, it is running on 120 VAC, and completely seperate from your lighting system.
No not really. Virtually all RV refers use 12V for the control board. But the refer 12V is likely from a different 12V fuse. Heat from the cooling comes from either 120V or propane but the refer will not run on either without 12V.
The OP stated that the lights go out and they are 12V. Good advice above on isolating the 12V problem.
OP When it goes out again check the voltage at the refer 12V. Be sure to use a good frame ground for all checks (do not use any cable/wire grounds) since the problem could be a ground problem to the refer.