Only 2 downsides come to mind
First, you won't have the automatic changeover to gas (unless you shut off the inverter) - not a big deal if you have a good memory or a checklist. I find myself sometimes forgetting to put the fridge back into auto-changeover when I've deliberately put it in manual mode for some reason. If you're boondocking just remember to manually put it into gas mode or else shut off the inverter.
Second, you've got to do a tiny amount of wiring to get the inverter AC up into the fridge compartment. You could install another receptacle with an "old works" box or just simply cut the male bits off an extension cord - thread it through a hole and put a new plug on. Not up to anyone's "code" but good enough for a test.