There are taxis and uber/lyft. I just looked at the bus schedules and I think you may be inaccurate about the buses no longer running after September 2; I believe they just mean the the summer schedule changes to a winter schedule starting September 3.
The schedule I have listed here, Route 10C, even says that it runs on Labor Day, most likely just with a different schedule (mostly that it doesn't stop at the lighthouse). (2nd column, 1st pdf page, 2/3 down)
I would check that page I just listed to see if it changes. If you are in Montauk, you can check at the initial Montauk bus stop. When you are there, call (on a weekday) 631.852.5200 and ask about it. Also on the day, the pdf file will possibly be replaced with the new one (same link, different file returned).
But you could always Uber/Lyft/Taxicab in an emergency.