It sounds as though you have a poor connection somewhere in the fridge circuit. The voltage at the fridge should not drop to 10V when the compressor is running; it should remain above 12V, assuming the battery is good and the voltage at the battery itself stays high.
I suggest you check all the connections, from the switch panel through to the fridge PCB, and clean/remake as necessary.
It looks as though the fridge itself is OK.
Good luck.
Roger