Drove the Narrows Bridge for years. I have no qualms about towing across it in the spring/summer. I'd echo the sentiment that you do not want to be in the far right lane going northbound. It exits very shortly past the bridge. The second to the right lane is the appropriate one for towing, but you will have to merge left or exit a little bit after the point where the carpool lane opens up to all traffic.
I prefer going US 101 to SR3 through Shelton, Belfair, to Bremerton. Nice easy drive, not very winding, and also quite scenic. But I'm usually coming from the south.