My first post.
I also just bought a Hi-Lo and love it. Enough on that for now.
To the point..I used to work on RVs for 3 years and then 15 years at an auto electrical shop.
Agreed, the converter should supply charge to battery when plugged into ac power. Ideally, when battery is CHARGED and on converter you should see around 13.x volts. Mine hangs around 13.4 which is enough to charge a 12v PB battery but not boil out the liquid.
To test for a parasitic drain you can do this
Find a Parasitic Battery Drain - wikiHow
but I personally dont. I use a 12 volt test light with a 1891 bulb ONLY. Take off neg cable. Put test light in series between terminal and cable and it should not light. If it does you have a substantial drain which, from your post, we know.
Now...unplug each fuse one at a time to find the offending circuit. Once located connect battery again and find out what is not working and drill into that circuit.
I would test for...
Fridge (may not be fused?)
Radio for correct wiring. These need a 12 volt feed all of the time (station and clock memory) AND an ignition (toggle sw). You may have both connected directly to to battery. BUT this should not drain the good battery in under a week.
Propane 12 volt valve if it has one. These will pull 1/2 amp which is substantial. This is a safety so dont remove it if the cause. Just turn it off when in storage.