Sorry buddy. Nope they will not do this during a regular service, I inquired at my AD and that was the answer after a phone call to Rolex London.
I've had work done by Roy S. Tompkins Jewellers 170/175 Watford market Charter Place Watford WD17 2RN He filled a dent I once put in a watch case and the repair was invisible.
Ah excellent, they're a stones throw from me as well. Thanks for sharing that I will give them a call.
If shipping to USA is an option, LA Watchworks are very good at what they do for laser welding. Highly regarded by collectors from all over the world. Shipping a watch for repair/return shouldn't be as hefty over the borders, if that's a concern.