I pulled the headset apart today and found that I did not need any measuring tools at all. Stamped on the underside of the crown race next to "Tange Seiki Levin" was the number "26.4". Similarly, when I removed one of the head tube races the number 30.2 was stamped on the side of it that was inside the head tube. Who knew?
I went ahead and calipered the parts anyway and the crown race bearing boss was 26.5mm and the head tube inside dimension 30.2mm.
Stack was 12.2mm for the lower bearing set and 31.8mm for the upper. Of the 31.8mm, 7.5mm is occupied by two washers and the front brake cable hanger. So that seems to indicate that the maximum stack height of the new headset can't exceed 36.5mm.
Now that I know my specs I'll start looking for a suitable replacement headset.