I finally figured out how to work the ending of the story. Or rather how to get it to work.
When Dr. Sturm changes the past, there is a time lag, as the changes propagate forward in time and outward in space. The further forward he jumps, immediately after making a change, the more likely he'll be in the future as the change sweeps through. And be changed by it. It never affects him—that he's noticed. But then he wouldn't, would he? He prefers to wait a bit and then jump forward.
So please do assume that I've hinted at and shown this effect a couple of places earlier.