If you have it turned down chances are there won't be enough meat left for a different thread.
You NEED to buy a pitch guage or at least offer up other threads to the stub to see if the threads on various pitch screws engage. They don't need to be the same diameter. You're just looking to use the other screws as a pitch guage. Try a variety of imperial number sizes and metric pitch screws to find one that has thread peaks that nestle in the thread valleys of your barrel for the whole length of the threaded stub. From there you can find the proper size.
Note that the major diameter of the thread which you give as .623 is not really going to tell you the actual thread size. Granted the .623 is preciously close to .625 which is 5/8". But threads are seldom so close thanks to the dressing off on the peaks of the threads. If I had to guess at a screw diameter from the .623 I'd likely try some metric screws or thread gauge sizes based on it being a 16mm size stub which is nominally 0.630". The .007 difference between the nominal 16mm to the measured .623 would be far more typical.
Regardless you won't be able to find that size tap or die at the local lumber yard or local Princess Auto. There's a world of special thread sizes out there in "engineering land" that are supported. But only by buying from such places as KBC Tools or Enco or other special tooling outlets.
And in fact when I go to the Enco website and look up "metric special" sizes there is a 16x0.75 mm size tap. So see if you can find a 0.75 mm pitch screw and see if it matches up.
Read post #5.
He will have enough meat if he threads it 9/16-24. It's a standard thread size for 10mm barrels. Add to that he will actually be able to buy a muzzle widget in that size, again, because it's common.
I would not have thought that it would work. But if that's the common size for 10mm barrels then fine.
What will happen is that he, or more correctly, the machinist doing the job will not be able to remove all of the existing threading without making the new threads too short. So the tops of some of the new threading will still have a rough look from the valleys of the old thread where they overlap. Lining up the engagement of the cut might or might not limit this by keeping the peaks of the new thread in the meat from the old. But at some point the threads will coincide and the new thread will have a dog chewed look on some of the job. It'll be slight for sure since the existing threads are pretty fine so they aren't all that deep. But it'll still have some mixing of the old with the new.