Bad news: setting the Z (apart from involving a bit of extra management if it did work) doesn't fix it unless I give it rather large differences in Z, which makes the sprites look really messed up and obviously misaligned with a perspective camera. It's kinda weird that it would do that. Regardless, having 5 to 15 spine skeletons will immediately look ridiculous if I give all of them unique, disparate Z values.
Anyway, there might be a better solution somehow. It doesn't happen with the goblins after all. I need to find out why.
This isn't a huge issue for now so I won't investigate it too thoroughly. But I'll be sure to let you know if something comes up. I'll just turn off double-buffering in the meantime.