That's pretty obvious, but your text is confusing me. You say:
The 5/8" lead screw shaft is 2" long, with a 1/4" step at the base, making the entire length 2.25". Extending the shaft by 3.188" would make it 5.188" with an overall length of 5.438". None of those numbers add up. You mention a 1" bore depth, which is of course the difference between 3.188 and 4.188. Are you saying the stub shaft needs to be 4.188", making the overall shaft length after inserting the stub shaft into the mill shaft 5.438"?