Just a little follow-up
I received a MT1 reamer today and used some grinding paste (from a lathe lapping kit) and just touched the surface with a reamer - just to test it. I did some lapping for 10 min and it is already very promising. The taper is already pretty good. I haven't tested it with a dial but before it has visual runout and now it does not!
When the "proper" lapping paste arrives, I wil give it a good spin (manually first) and then I am sure that the dial will show a perfect fit between the taper and chuck.
There must have been some burrs and other errors in the taper that lapping and 1 turn of the reamer has fixed. Well almost. Now I can't wait untill I can finish the repair and have a working tailstock.
Thank you so much to all for your comments and solutions. It really helped me to get going and to find and fix the problem. You guys are GREAT!