I have to ask to make sure!:::
How do you know the battery is fine? My battery this year, (one year old) was good enough to run lights, but wouldnt lift roof. it would seem to "charge to full" charge but when tested with an automotive draw (pretends to be a starter to the battery, which is what the pump would seem to be to a battery) it would dump to 4volts. couple bad cells.
Second, the shore power will NOT lift the top, by itself *USUALLY* (I guess if you have a 17ft size hilo? it might?) The shore power will assist a dying or weak battery, but wont help a dead/bad battery. Charger from DCconverter should trickle charge a battery... just a few amps, my guess is about 2 amps full charge then less as the battery gets full. Be sure to check water levels (I do it after a week's worth of camping).
I am assuming it doesnt even try to move (or even make a sound when not plugged in?) You might have more wires not connected at the battery? or the master switch in the wrong position, when not plugged in?
So you might have dead circuit, becuase battery not giving your lift switch power, but the inverter did, enough to run controls, when plugged in? When you lift or lower the top, the Battery is always involved, lowering the top with the inverter running and no battery, well I suppose it is possible since it can supply 12 volt power, but again I do NOT think it is possible lift without a battery.
FWIW, "Off" and lift only means everything except the fridge fan and fridge itself will be off (of course except the lift only),
The Fridge FAN and Fridge control items, are directly wired to the battery, so you are able to tow and not have fridge shut off when you shut the rest of the camper off... unless of course the camper's battery goes dead...
: 29ft '95 5th, converted to toybox in rear, by Hilo. Pics soon, Some Here