I have a Xantrex PROWATT SW 600 inverter. Constant run rating is 540W. While it does have a double-start watt-rating, generally that window is extremely short - like a second or two. I would be willing to bet my bottom dollar it won't start your fridge. At about 4.5A draw it just shuts off with a beep.
Otherwise it's a nice unit, easy to operate, the remote switch is convenient, and they're pretty cheap - I gave $150 from invertersrus.com via ebay.
My primary refrigeration is a Norcold gas-aborption RV fridge that runs on shorepower or LPG - this is a 2005 Winnebago.
I also just purchased a Bouge RV 12/120V compressor cooler/refrigerator/freezer. While it may not be quite big enough for your use, there are plenty of other options. The reason I mention this is that it runs on 60W fast-cool or 45W 'eco-mode' - 12V!
There are plenty of alternatives for this type of unit as well, larger, various configurations, some high dollar, some cheaper, and front-open styles too. Popular models from Dometic, ARB, Iceco, and a host of others - browse on amazon.
Another option would be a portable power unit, like a EcoFlow, Bluetti, Jackery, etc. I also just got a EF Delta Mini - it will power up appliances up to 1800W, with 882Wh of reserve power, or about 245A at 12V - my DIY lifepo pack is 230A, so it's pretty powerful and flexible, 23lbs, and about the size of a regular battery. Wife wanted to run her hair dryer-brush and the mini Mr. Coffee - neither will run on the Xantrex.
You could run your fridge off one of those and recharge it with your alternator. Plus use it for all kinds of other power needs.
Or just get a bigger inverter and cross your fingers.