I may be wrong but you said you leave the travel trailer plugged in (full time) at the house.. If thats right most converters unless they are a smart converter constantly charge a battery... Needless to say it sounds like the battery might have been overcharged (cooked) and that might be a big part of your problem... I would check the water levels in the battery first... Dry Cells kill batteries... and overcharging definately shortens the life of a good battery.... You might need to replace your current one... and only plug the trailer in for a day every couple of weeks or even better yet put an isolator on the batter so when you not using the trailer the battery won't discharge down to zero.... they sell one for marine applications that you can easily unscrew when you not using the trailer... Then you can use a regular battery charger or you can reconnect and use your trailers converter to do the job....
Half dozen methods... I have a little 40 dollar smart charger I bought at walmart that has a lighted scale that I use to keep my cars and rv batteries charge when its done charging the battery it has a green light to let me know its done..... Like a say there are lots of differnt methods... But constantly charging a battery without a smart charger will eventually kill your battery....
Mike C.