... The battery will die if I leave the doors off and don't drive it over a week...
Um... most cars will get a dead battery if you have them sit for a week with the dome light on...
If the "doors off" part is just a red herring you may need to check for drains. Or disconnect the battery if it's going to sit, which is what I do. $15 blade switches on all rarely-used vehicles makes it easy.