There are two possible issues. A Dorm type fridge is likely about 100 watts running and around 500 starting.. Note the starting power, that's the kicker, may even be more.. I know a 450 watt inverter won't kick start 'em.
The RV fridge 300-400 watts Running, and for that matter "Starting"
Note that the RV fridge DOES need battery power, and the converter is going to suck some power as well,, If the batteries are full up less than 100 watts. if they are "Hungry" Sizes vary but mine sucks a thousand.
Now.. If you have any kind of extension cord between the "Hole in the wall" (Wall outlet) and the RV.... Extension cords come in assorted sizes
12GA. up to 100 foot, but half that is better,, Should be OK.
14GA 100 foot you will loose too much voltage in "Transit" 50 foot is iffy 25 foot should be OK
16 Ga,, forget it,, Don't even try
I suspect this is the problem,, Too much wire between RV and house, not big enough.
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Use a 3-light tester on the outlets (you really should have one) to make sure they have power.. A NEON night light (Little reddish bulb if you look at the bulb you see two wires, no filiment, it glows) will go out if voltage drops too low, that said, the standard NE-2 lamp fires at 90 volts PEAK.. that's around 65-70 RMS.. which is very very low.. but if the voltage drops you should notice it (Dimms)