Ok, guys here is the report of my progress. After unsuccessfully calling couple bike shops around (some of them do not have ANY fork milling tools, and none had JIS), I decided to file it down.
I also discovered that race seat was a bit oval, 27.05mm in one direction, and 27.1 in another. After carefully filing it down a bit (more in the 'longer' dimension) I also cut a narrow strip of 100 grit sandpaper and polished ring a bit, getting rid of imperfections. After a number of 'try and repeat' iterations (took loooong time), race fit in just right. Of course, at that point, old race could be installed almost by hand (old one was about 0.05mm larger than new).

After that, I happily assembled headset, adjusted it and marveled at how smooth it is. Just to mention, as I was assembling it I just used a common sense and trial approach and found out that caged bearing 'wanted' to be assembled with the cage toward the race, and balls toward the cup; it felt better that way (stable, not rocking) and was spinning much better (when still being dry).

As I finished the project and was happily cleaning up my workspace I noticed the instructions that came with the headset. Hmmm - being still in excellent mood I decided to read them just for fun. Imagine how quickly my smile disappeared when I noticed that instructions directed to install caged bearing with the cage UP, and they were specific about it, both in text and pictures. Opposite of what I did. Darn!!!

After some thinking I decided that while I was fairly sure I have done things right, it was better to verify. Headset came apart again, bearings flipped, assembled back... as soon as I hand-tighten upper cup fork started to bind. After some playing with it, I decided that I was right, and instructions were wrong, and reassembled the headset my way, cage down, bearings up. Of course, bearings adjusted just right again, nice and smoooooooth.

Guys, what the hell is that?