I have a manually controlled three-way 3 cu.ft. Dometic fridge in my 2005 LXE.? When cooling using propane, the thermostat does indeed control the flame.? When I light the unit manually using the push button, it lights a pilot flame.? When the thermostat calls for cooling, then the pilot flame lights the full flame which heats the chimney, leading to a successful absorption cooling cycle.
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I have found that the propane powered cooling cycle is somewhat more efficient than the 120 vac electrically heated cycle, hence the same thermostat setting yields a few degrees cooler interior, which could lead one to infer that the cycle is continuous, but it is still regulated.
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Dave
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On a non electric controller, you light the flame by pushing the big button, and the fridge runs at a constant flame, and may get too cold if its cold outside.