One 12V battery should be enough to run the refer on propane for a week. There is something else running to drag down a fully charged battery overnight. Typically the furnace sucks 12V power if it is used. Add in a bunch of standard 12V lights and a dead battery will result.
A 2006 Montana probably has the IOTA DLS 55 converter. This is *NOT* a smart (multi-stage) converter. With 120V power (either shore power or an attached generator) it will take a long time (8 hours?) to fully charge an almost dead battery. If you are going to be dry camping at all, upgrade to a smart charge converter. One of these will re-charge a battery to 90% in a few hours (or less). The IOTA can be upgraded by adding an IQ4 module. It is a little plugin doodad and it works very well. Google for it and it should be in the $30 ballpark.
A 2005 Ford comes with the fuse and relay in place for the trailer charge circuit. Chevy/GMC do not. Maybe that is why a Ford costs more?
The suggestion to check the charge circuit is a good one. We see lots of folks with a blown fuse. The trailer charge line is typically a 20 amp circuit and will generally re-charge a battery (albeit slowly) as you drive down the road.