Back in 2006 I cobbled together an SQ-A system piece by piece and when I finally got enough stuff to attempt to use the camera, I had exactly that experience. In fact, that's when I wound up on APUG/Photrio!
I solved the problem by returning the back and purchasing one elsewhere. But I think
Zenzanon Fan's comment is the answer. Those cameras and their interlocks can cause a lot of head scratching! Somewhere on the Internet, and maybe in an old thread here, there were pictures of the internals of the backs. The mechanical complexity appears a bit intimidating, but probably do-able. Some removal of the leatherette is presumably necessary to get at things.
Ah -- easier to find than I thought ...
Fixing a Broken Film Transport
Good luck, between that and light seal issues, the backs seem to be the Achilles heel of these cameras.