Either way, interior trim has to come off, if the mirror has to come off too, it's only 3 screws.
This sounds like a similar issue to mine, since changing the door to another elite 03 door, there is no memory feed back from the mirror position as shown via tech 2 with TB. Mine is the same mirror that worked before, the same car loom obviously, so that only leaves the loom in the door.
A check would be, for me, to remove the original loom from my old door, and run it from the mirror plug, through the open window, to the a pillar plug, and see if the motor memory works. (then re set the air bag fault no doubt..? )
But the thought occurs, would it be possible to remove the working mirror and plug it into the non working side? Are the plugs the same? If the other mirror works too it's not the door or car loom, but the mirror itself at fault. No idea if the plugs and pin layouts are the same though...?