Having spent 9/10 of my life in NYC. Specifically the Grand Army Plaza area and commuting daily into lower Manhattan via Flatbush Ave into the Brooklyn Bridge, I would not want to pull a child trailer through that kind of traffic. After I met my now wife (blind) we bought a tandem together and we bought a Bob Yak (single wheel cargo trailer) and the fact that it was a single track trailer, and also the fact that anything inside it was either non-sentient or biologically inert made it a good option for us.

Here in Portland where the traffic density is not really that far off from Brooklyn's, I see a number of Dutch Bakfiets in use. The child(s) ride in front of the cyclist but they are still fairly low to the ground. Cars are cleaner than once ago, but trucks and buses aren't really. Being that Dutch bikes are manufactured in and imported from a First World Country means they are rather expensive. A new one is almost certainly out of the o.p.'s stated budget. I have no idea what their second hand valuation might be.

I may have missed the post that recommended some kind of Xtracycle wheelbase extension which would allow the mounting of a child seat and still provide carrying options for 'accoutrements'. That would be my recommendation.