There are 12v refrigerators available; check with RV and boating suppliers. I believe I have seen some that are dual-voltage (12/120).
If you go the 120V/convertor route you should be fine; a good=quality convertor won't burn up as long as it can produce the amperage needed to run the fridge.
Big rigs tend to run full-time, 24x7, so battery life shouldn't be an issue. The few times a rig is shut down are usually planned so the fridge can be emptied beforehand.