What would it take them to allocate a bike lane? No fancy construction
is needed, just make each car lane (there 4 lanes AFAIR) a foot
narrower and make a railguard to protect the cyclists.

Five lanes each way.

What would it take? A cosmic shift in the universe as we know it.

I suspect it would be politically easier for the folks who decide such
things to hang some bike lanes underneath or along the side of the
bridge.

It would be nice, though.