First, a tip on how to disassemble a piston QCTP ...
Take the body off the lathe. Rotate the handle until one of the pistons is as far out as it will go. Clamp a pair of long-nosed Vise-Grip type pliers on the edges of the extended piston, so it won't retract. Turn the handle until the other piston is fully outward. The handle and barrel can now be lifted out.
Secondly, a possible fix ...
Once you have the barrel free, unscrew the handle, take the barrel to the mill, and drill & tap three additional threaded holes to the beveled section, 90 degrees or so apart from each other and from the original handle hole.
Then reassemble the QCTP and install the handle in whichever hole best keeps the handle out of the way of other things.
PS - Hole locations and angles are not critical, and you might well be able to do this on a drill press. Obviously, esthetics and pride of workmanship play a part, but if you're a beginner (like I was), it's nice to know you have a bit of leeway. After all, you'll only be using one of these holes at a time! I did this to a number of AXA QCTPs that I was setting up for use on rose engines several years ago.