Bird successfully hunting bat on the ocean shore - Northeast U.S.

Was at Gooseberry Island in Westport, Massachusetts, United States and on the beach saw a very small bird chasing a very small bat at 1pm. I saw it successfully catch the bat through binoculars which was a treat. Bird had a fanned tail but was against the sun so I couldn't get coloration. Observation tower nearby which I suspect hosts a bat colony. My best guess is either American Kestrel or Common nighthawk.