I am not familiar with that unit, but my experience with various propane/natural gas appliances over the years has convinced me that the two most common points of failure on all of them are crud/dead bugs in the burner assembly and the thermocouple that keeps the gas flowing after light up. Everything else is pretty much bullet-proof. It has to be from a strictly safety standpoint.
From your description of symptoms, I think you have a failed thermocouple. These are a designed point of failure, frankly, and the last one I bought cost $5 and took me 15 minutes to replace. Given the low cost and ease of replacement, I'd throw a new one in just because and see if that doesn't fix the problem.
EDIT: Btw, no need to try to find the exact thermocouple specific to your fridge. You can buy a generic one at Lowes, Home Depot, Menards, etc. That's where I get them.