Lots of unknowns going on:
- Somewhere around 10-11v is a fully discharged 12v battery. 6.3v is way beyond. You might be able to get it to work a bit but you've probably done significant damage letting it get so low. Even if you do get it to take a charge, don't expect it to do so for long.
- 13.6v is conveniently close to what the converter puts out while charging. A fully charged 12v battery should be around 12.7v with the charger off, so you likely weren't seeing if the battery was fully charged. You just saw the charging voltage.
- Cranking amps is not what you are after with the house bank. You want amp-hours.
- If the fridge is running on propane, it's drawing a negligible amount. Either the battery was already dead or you have something else drawing on the battery.
- Did you actually put a meter on the wires from the truck showing amps going into the battery? Just because, it shows 12-13v doesn't mean many amps are going in (or it could be the same as when the converter is operating and the battery is simply not taking a charge).
What I would do:
- Get a clamp on multimeter (doesn't have to be high end but make sure it does DC current). Learn how to measure volts and amps with it.
- Start with everything disconnected and off (preferably off for an hour or more).
- Check the battery voltage. (if below 12.0v, you are starting with a low battery...around 40% of full)
- Check if there is any amp draw off the battery. (if more than an amp, you have something beyond the propane sensor...track down what it is).
- Plug in and get power to the converter.
- Check the voltage and amperage to the battery. Voltage should be at least 13.6v if the battery wasn't full and possibly over 14v if the charger goes into bulk charging. You should be seeing significant amperage (check the rating on the converter but I would say at least 1/3 to 1/2 the rated amperage up to the rating). If it's only putting in an amp or two, you have a charging issue to track down.
- If the voltage is up and it's putting in significant amps, let it continue for at least a couple hours.
- Check the voltage after a couple hours. It may drop back to around 13.1v if the charger is in float mode or it may drop back to 12.7 if fully charged (probably not but possible)
- Turn off the converter, let sit for 5-10minutes (the battery will gradually drop back when the charging voltage is removed) and check the voltage. Ideally, you should be around 12.5-12.7v. If it's below 12.0v, it's not taking a charge.
- You can repeat the tests hooking the truck up to see if it's really charging off the truck.
- If the battery is showing fully charged, disconnect the leads to the battery (don't trust the disconnect switch), let sit for the night and check the voltage. The drop in voltage overnight should be negligible. If in the morning it's 6.3v, it can only be a bad battery.
Possible issues:
- Since you report 13.6v on the charger, the truck shows a connection and the slide worked once you plugged in, probably not the charging system (do run the checks though as it may be something unexpected).
- The battery is my best guess but if the cranking amps are good, that's odd. Usually that would suffer on an abused battery.
- You have a 12v load that you are not aware of. You need to track down what it is and why it is staying on.
Try to be systematic and return here with the results and people can probably give you better feedback.