There is something exciting sailing out of a large iconic city. I wouldn't have missed sailing out of Vancouver in favor of dinner. I certainly wouldn't want to miss NYC. It is a bit more daunting when faced with the extended time to get out of port and to open ocean. I think as a human we are drawn to watching things going by while sailing.
For me, I would let the head server know that you will not be arriving to the dining room until after the ship passes under the bridge. Personally, on the first night, it is chaos anyway and the first meal is at a quicker pace as they are trying to get everyone in. Most people aren't seated until about 10-15 minutes after. Others are just late for whatever reason. They'll still work to serve you a full meal regardless of when you show up. As long as your are apologetic, sincere, and willing to tip a few extra bucks at the end of the week, no big deal.