I've been to Liberty Harbor twice and loved it. My suggestion is to just do it. I don't think the park will be open for long. I see development all around the property and I think they plan to sell out to developers in the not too distant future. You may never have the opportunity to camp so close to NYC in your lifetime.
The last time I was there, the entrance ramp to the highway was closed and I had to drive a few miles through tight city streets to get back on the highway. That was very scary and way too close for comfort a couple of times. I suggest you call ahead and ask them what the widest path in is. Be sure to check Google Maps for street closures and don't arrive or leave close to rush hour.