I assume this is automatically turning on at home with no other campers around. I remember reading a few reports of this happening when people were camping. Turned out for them- the remotes were RF remotes instead of IR remotes and were being affected by other campers (or something like that).
You've already touched on the other possibility- that there is an alarm. Apparently our kids are quite adept at setting it.
Not sure what else to tell you to check.
As for the dead battery- if you only have 1 battery in the camper, there are usually enough parasitic loads on it (things that draw a little all of the time- propane alarm, clock for the radio, etc.) that it will drain a battery usually in less than a week. Many new campers are coming with battery disconnects; the rest of us have to install them separately. This allows you to turn off the battery when you know you're going to leave it sit for any length of time (mine is- if it is going to sit for more than a day, I turn it off).