For the 4 years I owned my Repco 35 litre fridge, I used to set it up outside about 3 metres from my campertrailer. I ran it on gas at every campsite - very efficient, beer froze overnight, 2 kg bottle lasted at least 5 days. I never ever had a second thought about security. Most people at campsites have these in full view. Never heard of a theft. Perhaps because of the weight - have you lifted one full of food and drink? Be aware that in Vic these can't be used on gas on Total Fire Ban days, and can't be used fullstop in National Parks in S.A. Also, the Repco is not efficient on 12V.
I've now switched to a 50 litre 4 day cooler (not a deluxe $320 EvaKool, just a basic $140 "icebox"). Can stay in the car, or be put inside my campertrailer. Or hidden somewhere outside by covering it over with something. The record so far is 5-6 days on one bag of ice at an average ambient temp of 30 degrees.
In your particular situation, I would seriously consider as an alternative a good quality cooler - totally safe, and can be totally out of view in your tent. Just need to get some ice every 4-5 days.