Crank Bros headsets have the lowest stack height for an external cup headset that I am aware of. You probably want the stainless version as I’ve heard that the regular ones rust quickly.
Ragley Stubbing stem as suggested above.
I added a Works Components 1 degree headset to my bike. It had a larger stack height than the original hheadset and as I had the stem slammed, there were no spacers to play with either. I’ll bet my steerer is about 7 mm below the top of the stem. I did use carbon paste to help the grip a little. I haven’t died yet…
HTH.