How about if you have four days and don't want to drive too much:
Day 1: Maidstone & Sebonack
Day 2: Shinnecock & NGLA
Day 3: Westhampton & Friars Head
Day 4: Drive to tip of north fork, rent boat, cruise to Fishers Island, play 36.
I haven't seen it, but based on the news of what is happening at Southampton Golf Club I'd add it to the list and drop Maidstone.
On the same trip but on the public side how about Montauk Downs? You would also have to drive by the Lobster Roll a.k.a "Lunch" in Amagansett where any sane person would be inclined to stop.
You could certainly add a fifth day and not have to drive very far and still play and eat well.
Heck, you could go for 8 days, never leave LI (except for the excursion to Fishers), play 36 every day but only 18 at any particular course, and still be playing incredible courses -- add to the list abouve Garden City, Creek Club, Piping Rock, Atlantic, Deepdale, The Bridge, post-Doak North Shore, etc.