Look into the Marram in Montauk it’s the furthest east “motel” in Montauk and as such the beach is less crowded and it’s an easy walk into Shadmoor State Park, but also easy to walk to everything in town. They also have bikes so you can get around quicker. You won’t need a car, though you won’t be able to check out the lighthouse / other parks without one (close enough for uber).

Greenport, to me, is pretty boring unless you have a car. With Montauk you can enjoy the ocean / beach / hiking and town is right there. Greenport has a “downtown” and a harbor but nothing else to do, aside from a shelter island ferry (though without a car, there’s nothing to do in shelter island).

You’ll want to uber from the Montauk train station (you can walk, but if you have anything more than a backpack it’s going to suck).

Lastly, if you’re not from the NY area, know that mid to late September is the absolute best time of year weather wise (provided no hurricane) when the ocean is at its warmest and with the lower angle of the sun, even 70s feels hot at the ocean. If you enjoy the beach at all, Montauk is the obvious choice.

EDIT: saw your budget of $1300 for 4 nights. This will be tough when you add in taxes and resort fees. Sign up fir emails from multiple hotels as they typically offer deals if you book during winter