A case against one-arm handstand pushups
I would definitely discourage even thinking about a one-arm handstand pushup as something that goes along with a one-arm pullup. It's not on the same level. Not even close.
As a climber, the one-arm pullup is a staple. It's not the case that every climber can do it, but it's a common "feat of strength" that a lot of intermediate climbers train for as a side goal.
A one-arm handstand pushups is... well, I've never seen one. I can't even imagine it. Balancing on one arm is hard enough for even a seasoned gymnast, but then you bend it? I don't understand how you're supposed to maintain balance there.
I understand the idea of trying to match the one-arm pullup with its antagonist, and landing on the one-arm handstand pushup. From the perspective of "opposite muscles" it makes sense, but these are wildly different in intensity. Pulling your bodyweight is much easier than pressing it overhead.
I would say that the one-arm pullup is only slightly more advanced than a regular handstand pushup.
A case for regular handstand pushups (at some point)
All this is why overhead pressing should start at low-to-moderate weights, and increase slowly. Shoulders are weak by default, and shouldn't be subjected to your bodyweight as soon as you want to train shoulders.
I know you're trying to get at this calisthenically. Normally, I'd insist on dumbbells so you can increase the load as you progress, but in this case, I would recommend that you do pushups, but with your feet on increasingly high platforms. This will allow your shoulders and wrists to ease into the load of a handstand pushup via progressive overloading.
You could mix it with assisted handstand pushups. Rubber bands for instance. I'm having trouble coming up with how you'd attach to the bands, but I'm sure it's possible.