Turn everything off. Disconnect battery cable. Reconnect it. Does the battery "arc" or spark when you re-connect it? If so, you are drawing a load somewhere that you don't know about. That may be drawing you down pre-maturely… The bigger the arc, the bigger the load! Also, if you are plugged into 120V AC, and the AC stuff still works fine, and the battery keeps going soft, than CLEARLY the battery charger built into the inverter isn't getting 12v DC to the battery. Could be a fuse or breaker on/in the inverter…
2014 RAM 2500 Laramie 4x4 Quad Cab w/Cummins Turbo Diesel, 2009 Rockwood 2607, and 3 'yaks!
300W of Suntech Solar, a Rogue MPT-3024 Controller, and a Xantrex PW2000 Inverter