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I am a structural engineer myself, trust me when I say there are ways to do this. You could place the cycle lane in the middle and simply have it sheltered by the traffic either side, the bridge itself creates sufficient turbulence that you wouldn't feel that strong lateral blow. Anyway, there are examples of longer bridges with cycle lanes (Golden Gate bridge), it is a matter of cost as you say, but once you're doing a bridge anyway, factoring in the cycle lane is negligible as additional cost (plus the current cycle lane takes you almost to the bridge anyway). Also, if you compare it to running a 24/7 shuttle service for 100/150y (design life span of the bridge), you probably break even (not accounting for the carbon footprint of the van going back and forth, often empty for one leg of the journey)
It seems to be a case of 'be careful what you wish for'--the bridge can be subject to very high winds and I understand that wind protection for cyclists could cause high wind load from the side. A structural engineer once told me this about a possible Thames Gateway Bridge, and I've always assumed it applies generally to very high bridges, but happy to stand corrected if not. Obviously, another reason why it wasn't done (and perhaps also the decisive factor in deciding against making the bridge more proof to high winds) was cost, e.g. on-ramps and off-ramps and width.