Further, when such a system was (say) 90% complete, the other side would be in a use-it-or-lose-it situation. Fire at the US while it still can or accept US domination.
All the more reason to develop such a shield covertly; which I am sure has actually been occurring.
To add to this hypothetical question: What if the US announced the development of this SDI system, but also agreed to dispose of its own nuclear arsenal at the same time? Essentially building its shield, but surrendering its sword. Assuming the US actually held to such a bargain, would that do anything to assuage the world's fears of this being an attempt by the US to dominate the world?