when you say 1" is it a true 1" headset , as the 1" headsets are smaller than what is considered a bmx headset , even then there is two? sizes for those , 1" was used on smaller expert and mini frames , also road bikes,
you could shim the new race to fit the the crown on the forks, done this a few times , shimmed the even smaller French size crowns with no problems , bean tins work great for this , a bit more robust than beer tins , used an old stainless pan for the last one i did , which was the French size crown?
iso head tube with jis fork crown