new to the forum, but I tried to search before I ask questions that people have already answered over and over again, plus this relates to the topic. Is there a crossing of the Hudson closer to Manhatten?
This NYC DOT page has information on bridge crossings:
It looks like the Bear Mountain Bridge is run by a different organization. It has a walkway so it seems logical that you could cross on your bike even if you had to walk it. There is contact information at the bottom of the page below:
I hope this helps.