The fastest and most reliable way to store a battery is with a tender. Even disconnected internal cell Resistance will pull a battery down over time. What UAHaerospace said is very true.
If you have a stereo in the trailer I would bet the memory is hooked directly into the Bat+
You can buy an amp meter and hook it up inline to see what the draw is with everything off.
This is just an example, and be careful to read the max load specs but I have one of these for just the purpose of looking for less then an amp draws at rest.
Here is how this works. Replace the wire ends with alligator or battery clamps. Remove either of the battery clamps and hook the meter in between the battery and the load i.e one clip will be on the battery pole and one on the battery clamp. Any load will be shown on the meter. REMEMBER, DC circuits are polarity sensitive so if you hook it up backwards it may show a charge instead of a draw or vice versa.