I think it's up to you.
Hi Phil,
I think it's your choice. It's your watch, you are free to get it serviced where you like.
If you feel comfortable using your Local AD, use them. It maybe a lot quicker & easier for you. personal contact is always a good thing. And building up a relationship with your local AD is not bad.
I have used Bienne for one of my vintage watches & was more than happy with the results & price. The turn around was a bit long, but I have to admit a large proprtion of that was of my own doing.
If you want to use Bienne, you have to ship your watch there ( risk of loss/insurance ), sort out the paperwork, for when the watch returns to prove you own it & therefore are not liable foe imjport duty on the value of the watch, just the repairs done to it.
I don't think there is a hard & fast rule. I'm sure some of our US posters here can also recommend some othere watchmakers/AD where they have had good results.
I live in Europe & so would chose Bienne. (or my local guy on a little old German back street )
If you decide to go the Bienne route, I'd recommend you read Bill C's webpage describing how he did it. It's a few years old now, but details all you need to know.
I hope this helps.
Steve