From the description
@Huss gives, it ought to be possible to install a piece of foam or similar and block the light from the RF getting into the sprocket. Unfortunately, removing the top cover is a significant job because of all the functions on and under the advance knob. And, as you note, there aren't many shops outside Russia that will work on these FSU cameras.
FWIW, I have two Kiev 4 bodies and neither of them does this (one overlaps frames more or less randomly, the other has a slow shutter, but they don't do this). After attempting to jump to Contax, I've concluded that the most cost effective way to get a good camera in the Contax mount is to keep buying Kiev 2, 3, and 4(various versions) bodies until I get a good one. Compared to buying and repairing a Contax, this is likely to be many hundreds of dollars cheaper, and multiple years faster.
You may be able to trade off money for time by contacting one of the Russian camera repairers who do work on Kiev bodies, and inquire about buying a freshly repaired/serviced one. I'm about to do that. I like the system and the ergonomics of the cameras (not to mention the fantastic precision of the wide base rangefinder) -- but I just can't afford to play with real Contax.