When I was a young man, those two Indian places right next to each other, each about 6 feet wide, on or near St. Marks, were BYO. I was never charged a corkage fee.
I've never been to any of these places but thought this list might be helpful: https://www.thrillist.com/drink/new-york/byob-restaurants-in-nyc
I've done wine events at Peking Duck House in Midtown. Might want to bring your own glassware though.
Here's another list. It might be good to confirm as this might not be completely up to date (King Bee has closed). Have fun. http://www.wineberserkers.com/forum/viewtopic.php?f=22&t=23092
Ended up going to Ponche Taqueria and Cantina on 49th street. Small place, food was solid. Wasn't sure on the pricing, but we opened two bombers and one can and they charged us $5. When I originally called I think they said $5 for a bottle and $5 for a six pack, not really sure. But turned out to be a great place to do a two man bottle share and eat some good food.
