I'm not sure that I'm into (not that it really matters what I think) the notion of a span for "local traffic". Even with the current bridges, I can't imagine that on a "local level", either side has much to do with each other.
I'm pretty sure I've opined on this before (possibly in a different thread), but it seems that as much as just having 5 lanes (total) capacity, the fact that neither span has really any shoulders to speak of is a major problem. The bridge can be intimidating enough to some just due it it's nature, but when you're right against the barrier, I think it causes a lot more slow driving than if there was more space between the lanes and the sides of the bridge.
My thoughts are to just build a 3 lane span that actually has full left/right shoulders and used for the direction of travel that makes sense for whether it gets built north or south of the existing spans - Then convert the original 2-lane span to a 1 lane span with shoulders, and the existing 3 lane to 2 lanes with shoulders for the other direction of travel.
That said.... I'll be mostly surprised (but not totally shocked) if a new span gets built across the bay anywhere before I die.