Anyway, I made some other grafts afterwards, and they're all doing great now. I was really surprised how well two of them have done so far! For scale, the Pereskiopsis are approximately a centimeter or so in diameter.
Austrocylindropuntia sublata monstrose grafted onto Pereskiopsis spathulata. The picture was taken about a week after grafting and the graft was made on September 7th. Didn't sterilize anything since I was at my university greenhouse and didn't have anything more than old razors there.
Here's the other graft I made on the same day as the previous one, with the picture taken about a week after grafting. This one's a Pereskiopsis apical bud grafted onto a slice of the A. sublata monstrose grafted onto the Pereskiopsis rootstock. I wanted to see if the monstrose would get passed onto the Pereskiopsis, as it had for someone else on the forums.
And here's both grafts on October 7th, 4 weeks after grafting, and 3 weeks after taking the above pictures.
The scion is starting to show that sectioning that is present on mature A. sublata.
Not showing any monstrose growth yet!
The A. sublata scion came from this plant,
but it's starting to look more like a mixture between that and the old one we have at the greenhouse! 
I wasn't raised a Cactolic but converted to Cactolicism later in life 
