How old is the battery? How old is the propane/CO monitor? They do have a date stamped on the back of the unit. Sounds like your battery cannot hold a charge since they start squawking at around 11 volts.
As for how long, it all depends on the battery or batteries. Being a popup, that is about the only drain on the battery when parked and closed up. I can easily go a month without plugging in, I have done it a few times and still have over 60% left in the batteries (per Trimetric meter). I do have two 27 DC batteries. So yes, some of us little popups do have some decent battery capacity.
As for the OP, who knows? NO info given on trailer, battery, etc.