Reading your predicament of trying to restore a fine time piece, I can understand that it is difficult to find a reliable company that has the knowledge and access to parts to do a good job. The Good ones, as you have found out, are booked up in advance with a long waiting line. (This speaks for their quality work they do).

However all is not lost. Sometime ago I had the need to have a Angelus Wrist Alarm watch overhauled. After contacting several Watch makers, I finally found two companies which have very good repair facilities. I used Universal Watch Repair for the Angelus overhaul. The repair took a month and when I received the watch back, I was very much surprised how well it turned out. She runs perfectly, Alarm function works very well, the case was re-plated (20 micron Gold), new Crystal, all I all perfect, like new.
I also had the opportunity to talk to the watchmaker that die the overhaul and he gave me the current Value of this time piece at recent Auctions. So overall a very pleasant experience .
I certainly can recommend this company , they list Tag Heuer also , so check out their website, talk with them . I have no experience with Chronodeco but from what I understand it is a good company as well. Check them out . Let us know of your findings. Appreciate the feedback.

Here the websites: www.universalwatchrepair.net and www.chronodeco.com Good luck with your project, Heinz Rengel.