I believe refrigerators are still an option. Our 2007 25 did not originally have a refrigerator.
I agree don't try and run a 110 V ref. off an inverter. There are many 12 volt compressor or 12V DC and 120 V AC compressor refer.
You would need a transfer switch if you were to make the one outlet inverter and mains power. Please be very careful, if you don't fully understand the use of 120 V and use of an inverter. Mixing the two without proper switching can be catastrophic.
There are 2 types of refrigeration systems used in the RV world. That is an absorption type of refrigeration, which uses a heater and ammonia gas. These can be 120 V, 12 v and propane. These are NOT suitable to use on a boat.
There are compressor type, and that is what you should be using. There are places to use an inverter, such as powering an induction burner, or microwave. But these take larger inverters and large battery banks.
I realize that it seems one can save some money by using a small 120 V AC type of refrigerator. In the long run it is more expensive, because of the battery power needed.