I think trying to heat it or use a mallet on it is a recipe for disaster. So many things can go wrong that I'm not going to take the time to enumerate them. If you do, your next question will probably be: Where is a good place to go to buy a new copper sink to replace the one I damaged. I'm not unsympathetic, but that little residual amount of water is not going to hurt anything. Sponge it or wipe it when you are done using it, and learn to live with this minor flaw. I have a porcelain sink in my powder room, and the chrome ring that holds the drain in place sits a tad high, so I have the same problem as you do with water accumulation. It bothered me for a couple of days, and now I don't even notice.

I agree with kudzu,this isn't something to tinker with. Let a pro (not a plumber) handle it if you must. Changing the shape is done by a method used on old vintage car fenders called "srinking". Srinking is done with a hammer AND DOLLY. If you don't find a compitent tradsman,call around to old school car restoration shops. They may require you have sink removed before working on it. Those hammer marks are from origonal shaping.

Agree,..ask around if someone can modify it, if not, let it be.