I might continue being wrong, but:
Switching off the inverter can cause you trouble during power cuts. So it's not advisable.
www.payperwatt.com
So first of all, you cannot compare inverter batteries and AA batteries, as the structure is different (please refer to the table in the Wikipedia article).
For gel batteries, found the below, so it should draw less than with lead-acid one:
www.sciencedirect.com
As said, we don't have an inverter at this time, but when we had, as far as I recall, even with new batteries, it behaved exactly like you're describing... Of course, that was years ago so technology might have changed...