So ... Here's the outcome. Taking the advice of all, I came up with a collective solution.
1) I cut off the the stud flush with the back of the door using a dremel and cut-off wheel.
2) I drilled out the remainder of the stud using a bit the same size as the stud.
3) I located a screw with the correct thread (#8x32). This happened to be a screw with a combination (flat/phillips) round head.
4) I used a drill bit the same diameter as the screw head to recess the stud hole.
5) After being satisfied with the test fitting, I roughed up the head with emery cloth and applied epoxy into the hole; then set the screw head in place.
6) After the epoxy cured, I cut off the excess screw threads to be the correct height as the original (1/4").
7) It's now ready for paint.

Thanks everyone for the suggestions and inspiration.