The fridge control circuitry operates on 12V power. The fridge cooling system operates either on propane (with a little burner) or a 120V heating element. (There are some three-way fridges that also have a 12V heating element, but they are not common and it doesn't sound like you have one.)
While there are variations in fridge controls, typically they are designed to switch automatically between 120V power and propane: using 120V power when it's available, and propane when it's not available. Often there's a switch or something to force it to use propane even when 120V power is present.
The check light generally means that the fridge tried to light the gas burner but it didn't light for some reason--either something is broken or maladjusted in the burner assembly, or there is no propane, or there's a propane valve shut off somewhere. (At least some Dometic fridges have a gas shutoff valve built into the burner assembly that's rather easy to overlook.)
The 120V power for the fridge generally comes from an outlet mounted in the fridge compartment and accessible from the outside by removing the side fridge vent cover. This outlet, since it's somewhat exposed to the outside and could reasonably get damp etc., is usually protected by the GFCI. It sounds as though your fridge heating element has gone bad and leaks power to ground, which trips the GFCI when it turns on. When the GFCI trips, the fridge no longer has 120V power, and so it tries to switch to propane. That fails and the check light comes on. The solution to that problem is to replace (or have replaced) the fridge heating element...or to use the fridge on propane only.
You don't want to bypass the GFCI to use the fridge on 120V power as it is; the current leakage is potentially dangerous, conceivably even deadly under the right (wrong?) circumstances.