Oh, that's too bad. I've asked Abel if he'll have me back, so to speak, but I'll need some luck.

Would it be advisable to try using the official TAG Heuer service department (presumably via Tourneau) for repairing vintage Heuers?

I wouldn't generally think so, but I came across an old post saying TAG Heuer service was "running low" on vintage parts, which I take to mean they at least know what they're doing... so long as they have parts in stock.