I cant remember the exact dimensional difference between 1" road threaded headset, and 1" bmx headset.
But i cant see a problem with having a couple of rings (shims) made, to make the cups fit the frame properly.
This type of thing was available to make 1 1/8" headsets fit 1 1/4" cannondale/klein mtb frames.
I have had to shim the lower race on to the fork leg with shim steel (essentially sheets of coke can material), with no probs.
A well fitting ring would not affect the structural integrity of the headset.