The original symptoms of the propane alarm beeping when the interior lights were turned on...
As has been mentioned the propane detector has a low voltage monitor/alarm trigger. The interior trailer lights are 12 vdc as is the propane detector. There is no backup internal battery for a propane detector. I've not seen a smoke detector which doesn't have a backup battery included.
The RV converter/charger voltage regulation is designed to have the battery connected. Even an old battery will help. The battery provides a reservoir to smooth out the changes in load.
Without a battery in the circuit every time the lights are switched on the resulting change in load causes the voltage to drop before the regulation can respond. The propane voltage monitor sees the voltage drop and triggers the alarm.
Back in the day before motorcycle 12 volt electronics became necessary and more sophisticated, some owners would run with a capacitor to substitute for a battery. The capacitor worked to smooth out voltage spikes and dips.
In the case of a travel trailer the battery is also used to supply power to the emergency electric breakaway brake circuit. For that reason and possible damage to the converter/charger from voltage spikes, I would not run without a battery in the circuit.
It sounds like the battery was installed with reversed polarity so the new battery didn't correct the situation.
Propane monitors are not very selective as to what flammable gas is being detected. I know for a fact that solvents, hair spray, aerosol insect spray, etc will trigger the alarm. Be certain there isn't some consumer product flammable gas triggering the alarm before just replacing it. A new unit may not solve the alarm going off problem.
A bit more info is here.