Update: With kitten in place on top of the hatch, the beasty is quite willing to come into my trap system. He's moving one square every 14 ticks, and I'm keeping him in play by locking and unlocking the hatch while I wait for a dwarf to pull the lever that will seal the drawbridge behind him. As soon as that's up, I can leave the hatch unlocked, remove the kitten, and admire my new zoo piece.
My system is relatively complicated. Allow me to show it to you.
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On the left is the top level. It has all the dorf-safe access points and the "ductwork" of the project. You can see the kitten sitting on the hatch in the bottom left. On the bottom level, you can see that each module has a 9x1x1 containment pod with a 2x1 raising drawbridge on one end and a hatch cover on the other. The pods also have hatches just inside the drawbridge, for reasons I'll explain in a moment.
On the top level, above the center of the pod, you can see a set of floor grates. The pumps next to them will eventually be supplied with water and magma for the purpose of obsidianizing the beasty if I find that to be necessary. The two drawbridges inside the pod can be raised to lock the beasty into those five squares, allowing me to flood it with the right amount of magma and then water. The second hatch, just inside the outside drawbridge, is to play the beasty back and forth inside those 5 relevant squares until the inner drawbridges raise.
There are four drawbridges quarantining the hatches and the grates, because if there's a viable path through the hatch to one of the grates, the beasty will just destroy the grate right over its head instead of pathing to the hatch first. Keeping it moving between the hatch and the drawbridge by locking and unlocking the hatch is necessary to controlling its motion until the pod is sealed.
On the bottom level, there's a bit of ductwork with a drawbridge to let the beasty out on the north side of each pod. That drawbridge doesn't currently lead anywhere except to another hatch, which is for dorf use only. Eventually that access will be used to dig a tunnel for the beasty to go where I want it to go.
The whole thing is tightly secured with drawbridges and hatches in abundance to prevent unnecessary !!fun!!. The lever system is a nightmare. I've only got this module complete so far; the other four are not needed at the moment. I'll likely wind up repeating the entire system for the first and second cavern layers.
Well, now back to playing the beasty. My finger will get tired of advancing one step by the time I'm through...