Beyond the venture to Bethpage, if you are really trying to stay close to the Upper West Side, the Bronx's courses are the closest. Van Cortlandt Park is the oldest muni in the Country and despite it's routing being ruined by highways, it has been looking a bit better in recent years. If you can get there on a weekday you will avoid the 6 hour round, but you really should consider taking a cart. I've walked all 18 many a time but it's not a layout conducive to hoofing it. It's definitely a course for relaxing with a few beers and taking in some of the local "atmosphere". At the 9th or 10th hole you might be lucky enough for one of the "locals" to offer you a bag of balls they've gathered from the woods. Just remember he'll take half of what they say it costs and sometimes you can pick your brand... No joke a bag of Titleists for ten bucks.
Split Rock is definitely a harder test and a better layout, though without the history. It was redesigned by Trent Jones Jr. a few years back but I never really loved this course. It's long and straight and though the hardest of the Bronx courses it's just a tad uninspiring to put up with considering how many mosquitos your going to have to avoid. However, at the same facility is Pelham's original 18: Pelham Bay Golf Course. This is not a hard track by any measure, however I really think this is the most beautiful off the Bronx Golf Courses. It's certainly the most walkable and the recent renovations have made the green complexes really a sight worth seeing. It's a fun place to be. The entire time you'll just keep thinking "I'm in the Bronx...?"
If you are driving and looking for a deal, you can definitely find an online discount for Mansion Ridge up in Monroe (bout an hour...) I think that's the closest Nicklaus but it's condo golf at it's best and boring forced carry to shallow green Nicklaus "Signature" golf at it's worst. It should be well maintained. If you find a good deal online it's worth it (Rack rate is absurd and it can be an achingly long round) but only if you then go find another course to play, West Point is close by for a fun classic round, Pound Ridge is the New Pete Dye in Westchester that opened to glowing reviews. Hudson Hills is the next best in Westchester. Patriot Hills is getting good talk but I haven't played that one yet. But if you can swing it I would definitely recommend the Links at Union Vale. It was built by guys who got tired of the NYC 6 Hour round and went up where they could build a "Linksy" course. This is a golfers course. The love is there and it shows.
Marine Park a long track a long way from anything but it's a Robert Trent Jones and the trees are few and far between. This course has so much potential it's disgusting. It's a huge swath of land for this city but it hasn't been properly cared for and it seems to be falling into a swamp. It's a hard golf course for all the wrong reasons. I usually favor it's closer and much more playable Brooklyn brother Dycker Beach which is a pretty decent municipal golf course, but it's hard to recommend either to a traveler on a limited time frame. I love Dycker but only because it's closest to where I live and on a weekday it can allow 18 Holes before I really have to get back to work.
Note, all of these courses are to be avoided on a weekend. Your here to visit family and that's exactly what you should do on weekends. Golfing anywhere near the city on a Saturday has no doubt made people give up the game.
I'd say play one or two courses in the Bronx if you can (Pelham and Split Rock is certainly the easiest 36) and then head north. Jersey and Long Island are golf destinations in and of themselves. It might be worth it to plan a trip where each could be really be the focus. You can knock out a lot of worthy public access gems in Westchester/Rockland/Orange/Putnam countries which I don't think have the same potential for a golf specific trip.
Just a City kid's perspective. Have fun and mind the goose sh >:(t.