As mentioned by other forum members, Patek would be the best palce to service.
1. It probably should not be sent back to Switzerland to be serviced but to the local regional service department, which might be quicker.
2. The issue could be more that what meets the eye. Patek is best equipped to service the watch, check other problems which the AD watchmaker might miss out or feel no obligation to provide that extra mile of service. E.g, light polish, or oiling, gasket change, tolerance testing.
3. AD watchmaker might in fact not just void the warranty, but make the issue worse. Furthermore, the is no argument with Patek in future if your watch require servicing or something goes wrong in future that it has been tampered with. Mind you, sometimes they can tell if the watch has been tampered/serviced by people other than Patek.
I do hope you make the right choice. Personally speaking, it is still Patek's responsibility, and their honour which is at stake. They would definitely try their best to fix the watch. However, your local AD watchmaker might not provide that standard of care regardless how highly regarded he is local or regionally.
Have fun wearing your soon to be fixed watch!
*think of the cost you fix it AFTER the warranty expires. Doesn't it seem you are really making your money worthwhile by asking Patek to fix the issue now?