×1) Cannot lock him in place using 2 mirrors, because killing the brain won't release him.

×2) Cannot kill the brain and then force the goblin onto the plate, since he won't follow a direct southeast path.

×3) Cannot push a mirror onto its plate.

×Your third assumption is incorrect. Spoiler:

×
Holds:
Nanfro Budkin's First Dungeon,
The Puzzle Catacombs
Graphical Mods:
Power Station (AE)

×I killed the brain from its SW, though I reckon NE would work too. Oh and goblins shouldn't be on their ortho squares.

×disoriented's first assumption is the incorrect one (at least the first part of it).

×Lead the goblins onto the squares that are diagonally adjacent to the orthosquares. Get them in that position when you're on the main diagonal leading out of the chamber.

If you do this in the NW, you don't need any mirrors and that may help you figure out how to set them up in the other chambers.

×The basic idea here is that you trap the goblins on the tiles that are diagonally adjacent to the orthosquares, using their preferences to make them Wait instead of taking one of the orthogonal tiles. The preference order is N, W, Wait, E, S, NW, NE, SW, SE; however, only the orthogonal directions and Wait are relevant to this puzzle.

The NW is the simplest illustration of the idea. Lead the goblin here:



If you're on a direct diagonal to the SE of the goblin, then the distance to you from the squares to its south or east is the same as the distance to you from the square it's standing on. Since Wait has higher priority than South or East, this means the goblin will stay in place. Once you leave the chamber, the shortest path to you will always go through the diagonal (unless you cut off the pathmap to yourself) so you don't have to deal with that goblin any more.

In the other chambers, the goblin can move North or West, which have higher priority than Wait. So you have to use mirrors to block those options.

×You can get all four goblins on the plates at once, but you can also cheat a little by putting a mirror next to one plate and pushing it yourself - you'll be able to reach it on the same turn that the goblins in the other quadrants reach their plates. This lets you kill one goblin and only worry about the other three.

×Use two mirrors, on the tiles E and S of the orthosquare.

If you're having trouble getting it out afterwards, the answer is to stand east of the brain when you kill it.

×I think I might have been wrong with what I said before, didn't actually try it. I remembered doing something similar before but maybe that was a previous version of the room.

However, you can use a mirror to break the pathmap by blocking the entrance to the chamber you're in, and then step on the fourth plate yourself.

×if you kill the brain from the east, but only from the east and no other position