Glad to see you're taking the advice not to try and resolve the crown issue yourself Jonathan, it could have ended in a costly mistake on such a valuable watch!
Any recommendation would depend on where you are, so you'll need to let us know where you're based.
There are a number of accounts of TAG-Heuer service on the forum - they look to be good jobs, but they do seem to like to go the full refurbishment route. That's what some people want, of course, but for me reluming etc is never the way to go. As with much of watch collecting, it's a case of each to their own, I suppose.
It sounds more like you're after a movement service and lubrication, which a good watchmaker should be able to do for you - there are a number of people here on the forum who could do that, and more who could probably recommend one, but we need to know where first!