To add to what
@horsesboy said, yes BGW is responsible for mowing, making sure trees don't block signs, ect. They are not responsible for anything to do with the sign itself.
It's no different than if you owned a corner lot and there was a stop sign on your property. Your responsible for the grass around it, that your trees don't block it, and that no vines climb it. However you are not responsible if wind knocks it down, rust happens, or someone vandalizes it. In my role as a township planner we actually delt with complaints often of "so and so doesn't maintain the children at play sign, there's rust, I've been telling them to replace the post for a year." We'd often be like "this is the first we're hearing of it so we will fix it." The irony was often those people were the same that would then say they don't want government on private property.
The best BGW could do is put up a billboard inside the easement and maintain that, but there could be other restrictions to not doing something like that. The best BGW can do here is go to VDOT with complaints, ask for the change, and offer to pay for it.