Help finding headset cup

[bub]

So I’m trying to put together my ‘57-‘59 Hawthorne frame. There was one cup in the frame when I receive it but I need to buy a whole headset or just a new cups that fit in the head tube. I had two on hand which didn’t fit. I’m having trouble finding the correct headset or even just the right cups. I’ll be running a 1” threaded BMX fork.
I’ve added some pictures of the chrome cup that was in the frame when I received it.
The outside where the bearing sits is 43.5 mm and the part that goes into the frame is 33 mm.

The black cup is a regular 1” BMX threaded headset but it’s too loose in the head tube.

Any help would be greatly appreciated!
The only cups I found that may work are below. Would they work with a 1” for though?
https://www.bicycledesigner.com/bike-parts/bicycle-bearings-and-cups/headset-cups-type-b.html

 

[bub]

Won't work because the fork crown race is for 1''.

Sunlite makes a economical headset for a 1'' steerer with a 32.5mm cup diameter.
What I do is take apart a feeler gauge View attachment 94698Wrap a piece around the cup to create a snug fit between the head tube and press it in. I keep rolls of thin brass strips in varying thickness for shims.

Thanks for the info! I thought about just shimming the black 32.5mm cup to make snug to fit the 33mm headset but I wasn’t sure what to use.