It seems unlikely to be a short. One would expect a short to draw enough current to blow a fuse in the tow vehicle before the battery is discharged much. If the fuse has been replaced with a too large one, it might not blow and instead the wire to the short could get hot and melt its insulation. This is assuming the battery is healthy and can supply considerable current. Since it seems neither of those is the case you could hook up and look for some appliance or light that has been left on. If that doesn't pan out, use a current meter and look for current that is being drawn. Wiggle wires here in hopes of triggering whatever is causing the intermittent short or load that is running the batteries down.
It's possible that the battery is being drawn down while you are traveling and not just while at a rest stop or similar. This could explain how a few minutes in a rest stop with some modest parasitic load could result in a run down battery. Though one would expect the alternator to keep the battery charged. Alternators can fail of course, but here we have two different tow vehicles similarly affected and that points to a problem in the trailer. I.e., a stuck brake light or running light or electric brake or similar.
A device like this can be very helpful when trouble shooting a 12 volt DC system.
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