Your converter may not be able to maintain the proper output voltage when subject to the compressor's starting current . If the load on a DC power supply is greater than its rating the power supply will regulate
the voltage to get the current down to a safe level .The battery
has no such regulation. As Mr. Miller suggested I would check the voltage at the refrigerator
when the compressor is trying to start. "No load voltage is meaningless in this situation" I would speculate that the voltage at the refrigerator
is enough to make the compressor hum but not enough to develop sufficient torque to get the compressor to turn/run.