If your refrigerator is operating properly when on utility power, and your inverter/battery system is keeping it running when disconnected from shore power, chances are there is nothing wrong with your refrigerator.
First, get a thermometer in it and rather than going by "feel" get some actual temperatures to compare.
It's not unusual for a refrigerator to be slightly warmer when food and beverages are added. This usually means more door openings as well as reduced circulation inside with contents blocking some air flow.
Best this is to reduce door openings as much as possible. Rather than getting one item at a time to prepare a meal, get all that's needed at one time and put it all back at once. Same for beverages. Each time the door is opened obviously cold air escapes and warm air enters so limiting openings can go a long way to maintaining cold.
For a thermometer I'd suggest one of these. Is a wireless unit that shows both freezer and refrigerator temps continuously without opening the door. I even use mine as a remote thermometer, bringing the display in my house and seeing what my TT's refrigerator is doing from 50' away (TT is on pad in back yard).
https://smile.amazon.com/gp/product/...?ie=UTF8&psc=1
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