Just tuned into this one, so I'll throw in my two cents. A dremel is well worth the investment. I almost always will remove the bracelet and lay flat. For a heavily scratched strap I'll use a heavy duty automotive polish (red paste type). I'll smear this over the watch band, then with a wet polishing wheel I'll go to town. This will generally get you to 90% very quickly. I'll then clean and apply a brass and silver cleaner. Again, with a damp polishing wheel, I go over the whole thing. Another cleaning with soapy water, and possibly a once over with a clean, dry wheel.
I pretty much dismantling as much as possible (Caseback, bezel, etc), especially on Breits as these polishes will mark the AR very quickly.