OK, I understand what you got going on but probably can't help.
I think the drive gear inside the transmission is steel and may be part of the rear shaft, I haven't had one apart in years.
You may want to just order a new gear and see if that will work, 19 to 20 teeth sounds about right.
The gear you have may be for a earlier Fordomatic and there not the same as what was used in 1957, I think that you need a gear with a right hand cut.
I would think that some of the venders that sell the gears would know what they fit. Have you called Jerry Miller restorations.
Good Luck