You could try your luck with Oleg Khalyavin of http://okvintagecamera.com/ - he is more than good and experienced with Leica, Sovkam RFs like FED, Zenit and the Kievs derived from Contax II, perhaps he is able to fix your Contax III.
If it is the ribbons and minor "unspecific stuff", he should be able to do it for less than 100USD incl. postage in both directions.
And there is a fella in the UK, Steve Ash, working on Contax and Kievs.
He has a "check list" - wether it is worth the try or not.