Wheel stud replacement with axle shaft still in place, possible?

Don't drill it out! You'd risk more damage by drilling it than by driving the stud out. It shouldn't take that much force to knock it out of the axle flange. Use a lug nut & some washers like others have said to draw the new stud in.

If you drill it & you're off center or drill too big the splines won't bite into the axle flange correctly & you'll be in a bigger mess if the stud strips out & spins in the flange.