is your frame and old POS American bike with Ashtabula (one-piece) cranks? or a really really really old MTB?
BMX headset cups have a larger diameter than "regular" 1" threaded headsets. Thus you'll need to keep your BMX headset. however, you might be able to swap the for out for a road fork since the fork steerer diameter and threading is the same. This would also require modification to the headset locknut. or a new locknut.

I wouldn't sand off 1mm off a regular 22.2mm quill.
just get a frame that fits.