Unfortunately, 1" is not 1" in this case.

https://www.sheldonbrown.com/cribsheet-headsets.html

My frame and fork are both JIS, meaning I'd have to ream my head tube to install most threadless headsets (which are ISO).

I'll probably bring my upper cup and bearing with me to the bike shop today to see if the threadless race in their headset will seat with the bearing and cup from my old headset. If that will work, then I should be able to use both of my existing cups, and I'd be home free.