Yes the propane detector is something you want to remove or wire in such a way it is not running while the trailer is stored.
You can put a switch inline with the 12 volt
supply to it and try to remember to turn it on when using the trailer. Not likely.
You could wire it to the output side of the frig. on switch so that it was only on while the frig is on.
Looking at the year of your trailer I would suggest you charge the batteries then turn the charger off overnight. Remove the ground wire on each battery and test the voltage on each battery. They should be the same voltage at that point. Any difference and the weaker battery will draw down the stronger one. If difference is detected replace both batteries