Personally I would go a bit Nick Cook on it and not replace the bridge until all drivers start obeying the laws of the road. Imagine how many potholes could be filled in Morningside for the cost of the new bridge.
More seriously though I think segregating the cycle lane would be a good start. Alternatively It leads to a shared use path so why not have shared use across the bridge too. If there really is so much foot traffic expected across the bridge that it would be an issue then there will be huge problems on the A90 path.
Doing this would also add a more obvious route for cyclists heading west.
I like Blueth's suggestion of a stop sign at the end of the southbound slip. Although if public transport really was taking priority over private vehicles then the stop sign would be for traffic coming off the bridge and buses heading up the slip road would not have to stop.