If I wanted to spend an overnight, where would be the most economical place closest to the Shack?

You might find a room after labor day, in any of the local hotels (all within walking distance to gig shack) Daunt's Albatross is probably the closest. The surf club, Atlantic Terrace, and a few motels. Good luck.

Montauk manor

You may have luck on Airbnb

This time of year good luck usually everything's sold out but try Expedia.