We use frozen water bottles to maintain the temperature after chilling it on shore power prior to leaving. With 4 or 6 frozen 16oz bottles, you'll get through a travel day easily without thawing out. Then, repeat when on shore power.
A separate inverter would work for the fridge; however, it would likely put a large drain on the battery(ies) with little gain.
2017 Forest River Surveyor 247BHDS...
...Tugged by a 2016 F250 XLT FX4 SuperCrew, Shortbed 6.2L, 4wd, 3.73...
...Joined with a Husky Centerline TS 800-1200lbs WDH