After feeding him the candy stick, I use the mirror but it doesn't cause him to go back in his box.

The game is super picky: Don't have the bear in the hand, or the drum, make sure you did everything....

I don't exactly remember what triggers that bug, but I know there is a way of making him not react to the candy cane, maybe you managed the same.

I dropped everything except the candy stick and the mirror just to be sure and that didn't fix it.

I watched a couple of videos to make sure and I'm pretty sure I've done everything. I got the coin and used it in the machine to get the candy, used the oil lamp to defeat the puppet, played the drums to march the enemies blocking the doorway into the chest, and put the vanity case down to get the candy stick.

He is reacting when I hold up the mirror, but the mirror is only held up for a second or so, and he just sort of stops bouncing around, rather than screaming and retreating into the box. The videos I've watched it's held up for much longer; I'm not sure if there's some way of holding it up longer that I'm missing, or if that's just an animated sequence that happens.

Yeah, basically to make it look for longer you need to place the candy cane in the counter in front of him and then, when he is reaching for it (not after or before) use the mirror. My explanation from before is that I remember there was something you could do that'll bug that out and you reach an unwinnable state, but that is some 90s memories and I was never able to reproduce it.