Is there a consistent and fast way to get the mutant stuck where we want him to be stuck? I have tried different strats and none worked consistently for me, mostly the Mutant gets stuck near the robe guy.

Just moving up to the top right corner should do the trick. If not, then I'm not really sure if there is. :/

If he gets stuck, do you actually need him to move?

I did some testing, and after doing what is supposed to work he gets stuck, yet if i run to the locker again he runs to where he should be. So i guess we found a solution.