Converter/ batt. issues again?

Hello all, Here are the specifics. I have a 05 2619. It has one deka 12v deep cycle battery. I also have a honda eu2000 gen. Last year I was hooked up to shorepower at camp and overnight had battery die and set off propane buzzers, ect. Had 5 month old baby, 3 am., didn't know what was happening at first. Ended up converter not charging trailer batt., dealer replaced converter under warranty. This weekend, went out first time of season. Had camper plugged in for night before leaving home to shore power, towed 3.5 hrs. That night I checked battery on inside panel--showed one "dot". Hooked up gen and ran about 2-3 hrs, showed all while running gen, but when gen turned off after 2-3 hrs only 2 "dots". Was worried about loosing furnace overnight( now 1 yr. old baby). Next day got hold of multi meter, measured batt. before any recharge--- showed @ 11.90v. Hooked up gen., checked output of converter charging-- never went above 12.7 volts. That night ran gen. 4-5 hrs, charged through converter, two "dots" on inside meter, got through night. Towed home, ran fridge 12v both trips, checked inside panel, showed full charge?? I was TOLD by dealer orig. supplied converter( spec says American Enterprises cs-6000xl) no longer used/replaced by Trailmanor and newer updated converter installed. Or they just popped one in from other stock unit , who knows? I don't know how to tell exactly what converter I have. I don't think it is putting out enough volts to charge batts., I also believe my battery had a poor life so far. Sorry for long post, wanted to be clear. Thanks for any help/ info.
Scott