I once had a problem with my fridge not cooling properly while towing. I called Dometic and they sent me a baffle kit to keep the wind from blowing out the flame. Evidently, Dometic is OK with having it on while towing.
I always tow with fridge on....
2016 Siverback 33IK
2015 Ford F-250 Lariat 6.7L V8 Diesel 4WD Super Cab
"If you don't know where you are going, any road will get you there."