There are some good reasons why the very few remaining Contax mechanics can do a good business repairing these old beasts.
They are extremely complex and non-intuitive. The cameras, that is, although it may apply to the mechanics as well. The problem of coordinating the actions of leading and trailing curtains was resolved by getting the friction of the shutter tapes correct. The original material is unobtainable, and there is no really good substitute. I had been using substitute material that was too thin. Amazingly the shutter curtain coordination seems to depend entirely on the static friction of the shutter tapes on the slots in the leading curtain through which they pass. My thinner material had insufficient friction, allowing the shutter to cap if the trailing curtain was released slightly ahead of the leading one. The correct adjustment appears to be to release the trailing curtain an instant early. It will not move due to the tapes' stiction.
The two curtains move simultaneously, and when the leading curtain stops at the bottom of the aperture, the trailing curtain continues, completing the exposure due to the momentum of the shutter roller and the fact that the tapes are experiencing dynamic, rather than static friction. This sound ridiculously fiddly, but it works, and is consistent. The shutter closure is aided by the fact that the mid and slow-speed escapements are disengaged by the incredibly complex gear train when the trailing curtain gets close to the bottom of the aperture. That is if all the tiny springs in the shutter mechanism are pulling just the right amount.
The material I ended up using is 3mm Kiev tape, which is not as smooth as the original Contax material, and is ~0.33mm thick. I find that is does the trick if the slots in the trailing curtain are very slightly relieved with a tiny jeweler's file, not to open them up, but to bevel the sharp edges so the rougher surface does not hang up.
Why did I subject myself to MANY hours of frustrating work for a camera which had many things wrong with it, and is not particularly valuable even when fixed, other than the obvious point that I was too foolish to give up when I should have? This is the best Contax III I have ever encountered. Cosmetically it is near perfect, and the meter not only works, but is accurate over its entire range, the only time I have ever seen this.
There is a moral to the story. Never undertake a repair of a difficult and complex mechanism until you are completely certain that you completely understand its operation, or be prepared to spend many frustrating hours while you learn the hard way.