I only have two coach batteries and my inverter is 300w. I suspect that isn't nearly enough to operate a fridge. What size would I need and aren't inverters battery "eaters"?
A typical modern residential fridge uses around 500 watts when running, but you would have to check the rating on yours. If it is given in amps, then amps x 120v will give you approximate watts. The inverter would have to be somewhat more than that to allow some margin, say +25% greater to be safe. So maybe 650 watts? Or better yet, 1000? Not much price difference anyway.
As for inverter being "battery eaters", no more or less so than any other high-amp usage. 500 watts @ 12v is about 42 amps, so yes it sucks amps fast. You would probably need more batteries if you needed to run very long on them without an other 12v source. However, when driving your engine alternator is providing 12v at a steady rate, so the batteries aren't a factor. The inverter, however, still has to be big enough to handle the wattage.