converter help needed

Yeah, tricky.
Assuming 100 Amp hour at the 20hr. rating, 50 AH being usable: This mini fridge is undoubtedly a 120V device so the draw at 12V will be 15 amps. The variables will be the duty cycle of the fridge, the inverter loss, and the Locked Rotor Draw. Most small fridges run at <90f will have a duty cycle of 1/4, i.e. the fridge will be running 1/4 of the time. The inverter loss is dependent on the quality of the inverter, cheaper ones having 10% or so loss, better ones having 4% loss. The locked rotor draw for my 1.7 cu. ft. fridge is 6 amps. It is of short duration but still adds in to the total. These would be the theoretical electrical variables. You also have to figure how many times the door is opened, available space for air transfer, and the contents volume. Another factor is the low voltage threshold of the inverter. At close to 50% DOD, your inverter will be pulling the voltage down below 12.2V and may trip if it exceeds the low voltage threshold.

I'm going to WAG that your mini fridge will run for about 6 hours, maybe less. The converter has nothing to do with the operation as it only is used when the trailer is powered by 120V.

How long your battery will last depends on the Amp Hour rating of your battery. For example, if your battery is rated at 100AH, 50% of that is available for use. If the mini fridge draws 1.5A at 12v it will last 50/1.5 hours. The converter doesn't matter, as beemerphile said.

This seems like very little power for a fridge. If the mini fridge is a 120vac fridge, you will need an inverter and the 100AH battery would last about 3 hours since the current at 12v would be about 15A.

Is this some trick question?

100 amp battery has 50 usable amps to avoid damage
1.5 amp fridge will pull 50 amps from battery in approximately 33 hours.

Converter has nothing to do with it unless you are plugged into shore power. If plugged in the answer would be indefinitely.