I think this one is solved.
I tested the runout on the chuck body and socket, and the arbor separately and they were both 'OK' 0.07mm (about 0.003")or less runout.
Mated them together and the 0.35mm runout was back, but wait....that didn't feel tight.....No matter how much I 'wrung' them together I could still feel looseness.
Just to confirm that I wasn't hallucinating, I put some Prussian Blue on the arbor and 'mated' it with the chuck socket.
It's not easy to see but the Blue at the wide end of the arbor isn't 'scrubbed off', and that's where it felt loose, too.
So while I was pondering this, and staring at the chuck and arbor I noticed: Chuck B16, Arbor MT2-JT33
I hadn't realized that B16 is actually a taper (it's the same as Morse 2); I'd thought that it was a chuck size designation of some sort. Definitely different than Jacobs JT33.
So arbor and chuck don't mate properly.
It seems to me that a new arbor - MT2-B16 should fix the problem. Does this make sense?