Drill the hole larger on one side of the new handle so it slips on. I'd put a drywall or similar small screw on the other side, cut off the head, so the handle slides onto the two screw 'studs'. Put some 5 minute epoxy in the handle holes, slide it over the studs, and wait 5 minutes....
Edit - or for the second stud use a screw in the handle, and put the epoxy in a drilled hole on the door.