I should have done a bit more reach on the best route to GWB from Jersey City but I cobbled together a route up Palisade Ave as far as I could go and then cut down (west) to River road then north to the GWB.
Dumb question, but what part of Manhattan are you going to? If it's in lower manhattan, the ferries across the Hudson from JC & Hoboken will let you bring a bike. (
Liberty Landing ferry, bike is free;
NY Waterway is $1 for the bike.) It might save you going all the way north to the GW and then south again in Manhattan.
If you're on Communipaw, you head up over the hill, back down, left on Grand and follow that to the water. Paulus Hook, which has a NY Waterway ferry dock and the Liberty Landing dock is along the water to your right (If you reach the Hudson on Grand, look right, the NY Waterway dock is right there
)