If I was a betting man I would have put money on heat-treating, but 10 minutes of red-heat didn't soften the weld all that much. (It was noticeably easier on the threading tool though. )

Grinding the weld off with the Dremel worked rather better, which suggests to me that it's only the visible weld ridge on this bit of pipe that's particularly hard.

Gordon W's suggestion would have been overkill in my case but I can well believe that his 'belts and braces' approach might be essential on other pipes. I'm learning that, when it comes to machining it, there are often huge differences between similar looking metals even when they have the same name.
Hopper and Martin both make very good points about seamless. Next time that's what I'll ask for. Another disadvantage of heat-treatment is that restoring the finish later is even more work. And weakening the seam of a pipe that's also been undermined by cutting a thread could well be problematic even if I'm not working for NASA!
Thanks again,
Dave