After spending most evenings before dinner tinkering, I think I've discovered the problem with multiplication. All 1s-4s transfer to the register while all 5s (and above) don't. I got 5s to work when pushing the carriage diagonally, so I think there is a carriage alignment issue.
I may need to pause this restoration for a bit. It is looking like I may need to replace my alternator in my pickup and I only want one disassembled machine lying around at a time.
The way this mechanism works, only some of the gears from the keyboard mesh with the multiplication register gears. Numbers 5 and above are too far out of alignment to mesh with that register. For some reason (angles I guess) this doesn't affect the upper gears as much.