Fridge has a 12V electronic control panel that is continuously powered when it is hooked up, whether switch is on or off. Mine says on side of the door that the DC rating is 1 amp and AC rating is 300 watts (Norcold). Your readings confirm that 1A DC draw when on propane. I've never checked the DC draw on mine when off, but I'm not surprised there is still a draw. Old propane fridges that didn't require 12V at all obviously don't draw any DC, these are the manual pilot light kind.
My guess is that the small single battery was severely discharged and damaged during storage. As you've discovered, there are parasitic draws on the 12V system even with everything shut off such as fridge and propane detector. As your trailer sat between trips (assuming it was unplugged), the battery was depleted to death. So despite how long the stock converter or tow vehicle had to recharge it, it is no longer capable of getting anywhere near full capacity due to damage.
I think solution is to replace the battery and get in the practice of storing it with a known full charge and then disconnecting it between trips. Either by installing a battery disconnect or disconnecting the negative battery post from ground. Or even taking the battery out. Solar works great, too, but it doesn't sound like you're an avid boondocker so investing in that may not be worth it.