There is more room at Shinnecock for sure but it's very hard to get a camping reservation there. Here is the
link to the county website. For all Suffolk County Parks you'll need a non-resident outer beach permit to drive on the beach and a non-resident green Key card to access the res system.
Suggestion would be to go out to
Montauk County Park, campsites are also limited there, but if you have a truck camper and a state fishing permit, they let you hang around Shagwong Point. You don't need a reservation and even if the CG is full they'll let you out to the point with a truck camper to stay over. Not sure if they have stay restrictions on the TC crowd at the point.
The beach is too narrow there to park a trailer but you can drive a truck camper there no problem. You'll also see some 4WD class C's. I would call the park before you go, we had a bad storm season this spring and some of the beaches got hit pretty hard.
There are also NY State beaches you can drive on but they don't let you stay overnite.
PM if I can help. This was us last Fall at Montauk (notice not very crowded LOL)