I guess I knew this and must have read it somewhere, because I wasn't totally surprised when I got my 2-1/4" Happ mounting plate from and found that it didn't fit on my 2-1/4" Betson Imperial trackball.
Problem 1a: the holes in the betson trackball case are not quite large enough for the posts in the plate
Problem 1b: the posts in the plate are slightly more 'spread out' than the holes in the trackball case.
I sanded the posts on the plate and my wico 2-1/4" TB squeezed on. Since I wanted to be able to remove and re-add, I used a 13/64" cordless drill bit, IIRC. Of course, you have a betson 2-1/4", so differences and all.
I bet your redrilling didn't center
absolutely perfectly with the original holes, especially if three fit at a time, but not four. Very common in re-drilling with normal bits, and more even common with multiple re-drilling.

I WAS surprised, however, when I found that my 3" Betson Imperial trackball didn't fit into the Happ 3" mounting plate. Why was I surprised? Because Happ's own description, of the plate I purchased, specifially says: "For use with both 3