After some measuring, it seems I need the bottom of the stem to minimum insertion line to be approximately 2.5 inches instead of the 3 inches that are pretty common on stems.

I have a old Nitto Periscopa (I think?!) laying around from a vintage mountain bike that is 2.5 inches and that seems to work fine. Are there any modern stems with only 2.5 inches that will get the bars up high enough?