I believe that eventually the existing bike lanes will be replaced with a bi-directional MUP on the north side of Winston, so the new bridge will eventually feed eastbound bike traffic onto that. Until that's complete (and I've no idea when that will be) there seems to be no option but to take the bridge and then re-cross Winston to access the eastbound bike lane on the south side of Winston as you describe. It would be much safer if direct access to the eastbound bike lane were available without having to cross the bridge.