Your symptoms sound a lot like mine. This is the same failure mode my WFCO converter experienced.
Open circuit voltage (voltage with the converter disconnected from everything) at the converter terminals would read 13.2, but it could provide very little current. With the battery connected, the battery could support the load which is why it took me a while to figure it out. Eventually the batteries slowly discharged. When I disconnected the battery, initally it looked like it was working. But I quickly discovered the converter could not provide enough current for much more than a led light or two, maybe a fridge control board. Once I exceeded about 1A current by turning on a couple other things it would crash. Lights would dim to almost nothing, fan would bairly rotate. I could shut everything off, unplug the converter and plug it back in and it would return to 13.2v, until I loaded it past 1A.
All WFCO converters are designed to provide power for 12V appliances and lights without batteries. It should be able to power every 12V item in your trailer and charge batteries at the same time. Your converter has failed, replace it. I recommend a Progressive dynamics PD9145c, can be easily found for under $150.