That is pretty clever Doug, will certainly keep that one in mind !
I have a friend who is very much into old RC engines, and faced the same challenge a couple of years back. He found that it with tiny tips like this, is much easier to "grind" them instead of cutting.
He uses his Dremel in the lathe, with one of these adapters:
https://www.ebay.com/itm/The-ORIGINAL-Dremel-tool-holder-frame-1-2-in-X-1-2-in-Shank-for-your-lathe/113612709777
Might not be of use, or even interest to you, now that you have found this solution to your immediate problem.
Just thought iŽd share, in case someone should face a similar problem, with something needing a slightly different angle than that of a phonograph needle.
That is interesting. I could see other possibilities for it as well!
crazydoug