Welcome to the club. We purchased a Montana High Country 9 days ago and are now on our second extended week-end camp out. As far as traveling, during the walk though it was explained as long as I'm hitched and running the battery will continue to trickle charge from the tow vehicle.
When returning home from the first week-end, I plugged the camper into shore power... but ... forgot to flip the breaker back on. It sat for several hours (like 10 hours) when I realized the breaker was off all that time. The refrigerator did just fine. The house lights were still bright as ever after all that time. The refrigerator never missed a beat.
So, I really don't know, but I'm pretty confident an overnight with no power will not cause the refrigerator to get warm. Now????? if the battery begins to get week and fail, age out, or just go Kaput! Well, that may be a different story.
Congrats on your new Montana!