Could the $74 guy be using an extension cord to your pedestal during the week while the park is empty? Really just kidding on that one but I couldn't help it. :)
Going off the electrical gremlin theory. Unplug the shore power cable and bleed any surface charge off the battery. Try to get the furnace blower running for about 3 minutes. If you can't, then turn on every 12v electrical appliance you can for three minutes. Think range hood vent, bathroom vent, overhead lights etc. After three minutes turn everything off and let the battery rest for a minute or two. Now take some readings.
Pay attention to battery voltage. What I mean by that is measure your voltage across the positive and negative posts on the battery and write it down.
Now anytime you measure for DC current whether at the fuse box or at the water heater board, you're looking for battery voltage. If the battery is at 12.67 volts, you should see 12.6x volts or at least close to 12.6x volts at the water heater. If you don't, then start going upstream until you find the voltage drop.
Measure the voltage at the furnace or the water heater with the appliance turned off. Then turn it on and measure again. Are the two readings close to the same?
Starting to think further, you might have a high resistance problem. A multimeter pulls almost no current. So it will measure 12 volts through a single hair thin piece of wire that would never support a furnace fan. If you have a big voltage drop check on the trailer tongue for a circuit breaker or fuse on the positive wire from the battery. If a circuit breaker make sure you have the same voltage on both posts on the breaker.
You can always jumper the two posts with something to rule out the breaker itself. If the breaker is bad they are available at any auto parts store and probably at Walmart as well. Just make sure you look at the amperage rating on the side of the old breaker as like fuses they come in various amperage's.
You can also check fuses by taking a measurement off back of the fuse. Poke the positive lead of your meter into the slots on the back of the fuse by the rating number. Those slots are at the end of the two blades on the fuse so you can take reading without unplugging. You should get the same reading on both slots. If the readings are different replace the fuse.
If you suspect a ground problem, or just want to rule out the ground you could make a temporary ground. Use a spool of wire, some jumper cables clipped end to end, a length of copper pipe and some alligator clip test leads. Something to give you a long return path to the battery. But if you use a fabricated ground, make sure you take your battery voltage test at the battery using the extended ground.
As far as your converter charger the easiest check is to plug in the shore power cable and do another battery voltage check. You should be getting around 13.6 volts at the battery. Again, you should be getting close to that same voltage at the fuse block, the furnace, or the water heater.
The high power bill was probably your charger running full bore trying to charge up the dead battery. If you suspect anything else then turn off and empty the refrigerator and bring the food home. Flip the main breaker in the trailer not at the post and disconnect all negative wires off the battery. Take a picture of the electrical meter and then head home. The next weekend take a look at the electrical meter when you return and make sure it's the same. Hook up the battery, flip the breaker back on and turn on the refrigerator. Reload the fridge with the pre-cooled items from home.