Hi,
In my opinion the answer is in the question! 
Speaking about the Stern agency I suppose you live in the USA. I'm not a PP collector so I have no clue about a place where you could send your watch to be restored, except Patek Philippe directly.
Unless someone here has another solution to propose (through private mail please as per forum rules) if I was the owner I would give it to the Stern agency for a quotation of restoration. It's certainly going to be expensive in absolute but probably not so much compared to the value of the watch if you want it to be restored.
Cheers
Dje