That's kind of the long way around, you could come down I-95 into I-278, Gowanus Expy in the Bronx the Brooklyn-Queens Expy (BQE) when you get on Long Island. Either way you wind up in te same spot, at the East end of the Verrazano Bridge. Honestly not sure about which local roads in Bklyn are better. I wouldn't worry too much with a C, but I would be extra careful pulling a trailer just because of tight turns.
Maybe see if you can get a local number for the CG and ask them. I grew up here and never knew you could camp there....
