×Specifically, how do I push him out of the enclosure?

It seems that we both have to enter, because the tar blob can only be removed with a simultaneous cut from two swords. Then, there is a special dance where Beethro's sword points up, the slayer's hook points down, and when Beethro goes south, he goes south too until he is two steps to the SE (or SW). With this, I can push him out in the south, to the spot below the trapdoor. The problem is, he won't stay there. When I exit to the north into the tunnel, he steps inside again, and he becomes trapped, because I can't help but strike the orb.

×When the path to you gets cut off, the slayer follows his wisp for as long as he can.

So resolve the problem by making sure his wisp doesn't take him across the arrows until you've hit the orb.