A shim between bottom headset cup and bearing
The custom shim works, I settled with a ~1.4mm option, this was the minimal thickness machinist agreed to make, he said "it would bend in your pocket otherwise". After properly preloading the headset bearings I did not notice any play during pedaling efforts or when applying front disc brake during ~200km the bike made since assembly. That's not much distance, so if anything happens and I survive I'll update the answer.

Apparently, some people experience the same problem. 

To get your own shims, find a local machinist and use the blueprint from the question, adjust thickness as needed. My shims were made from a 2024 aluminium alloy plate. The outer diameter tolerance should be in the ballpark of +0mm -0.03mm, the rest doesn't matter.