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Bee and thistle is excellent dining. Full of excellent artwork. Great booze too. Figure about $100 per person before tip depending on your wine tastes. Everything on the menu is outstanding. Fairly casual. Nice jeans, a tucked in polo, and a blazer / sports jacket works fine here. No need to get dressed up, but I'd advise against buffet attire.
It is right next to the Florence Griswold Museum, which is a wonderful, small art museum that is world renowned for it's heritage, collections, and ongoing contributions at the nearby art college.
Don't forget the USS Nautilus, the first Nuclear Powered submarine (and only one on display in the USA IIRC). The Submarine Force Museum and Library accompanies it. Admission is free IIRC.
http://www.ussnautilus.org
Battleship Cove in Fall River is like a 90 minute drive from Mystic. Not sure why the guys are sending you there.
Take a Ferry ride to Fishers Island. I know it's going back to NY, but you can't get there from NY, so that's kind of neat.
Octagon steakhouse is excellent as well. It's owned by the guy that owns the casino and pretty much any decent accommodations down there.