The battery runs the control board on the refer, water pump, lights, CO2 detector, furnace fan and control board, power supply for television antenna booster, and any built in entertainment (radio / CD player etc) system. It won't last all summer.
Charging with car and jumper cables, is not very effective and very costly. You're going to need some other type of charging. A generator and / or solar panels and multiple batteries, would be more in line with a seasonal use.
If the trailer is sitting unused and the battery is left connected, even with everything turned off, parasitic draws from the control boards, radio, and CO2 detector, will drain the battery in 7 - 10 days.
Using the furnace on a single battery without charging, you can estimate your run time in hours, not days.
