There may some very modern fridges that actually turn the gas on and off reigniting each time, but certainly most continue to burn as long as it's turned on, and the modulate the flame size according to demand, so continuing to burn is probably normal.
What is not clear is if the fridge is modulating, and if not why not. One simple reason could be the inside of the fridge is still warmer than the setting dialed in at the control so the control keeps the burner operating at high power.
Because the fridge still apparently works when connected to 230Vac, that confirms the refrigerant is still present and working, but not on gas, so this is beginning to suggest the burner may be damaged in some way.
I strongly reccomend you get the caravan checked by an approved caravan engineer who should be able to check and verify what the problem may be, and repair it.