Jackie, the route you propose sounds amazing!
I have crossed that bridge by car on my way to Petersburg historic sites and can tell you it lacks sufficient borders to pull over or sidewalks, forcing you to ride on the traffic lanes. You also have the steel grid plates at the middle of the bridge, a real hazard for narrower tires. VDOT and others have been talking about providing crossing alternatives at this point since at least 2017 (
www.craterpdc.org/environment/documents/Technical%20Document_071118.pdf
) such as a dedicated bike/pedestrian ferry or shuttle vehicle, neither of which have taken hold.
May I suggest your best option to do this route is post a request on the day of your ride for a lift across the bridge on the day of your ride. It would be a matter of coordinating the time to meet and where just before the bridge (there's a few options). Depending on the day and time I would be willing to help you out.