Can you spend three fatigues on telekenesis to suck a monster 2 squares away from you into your aura, back out and back in again?

Yes. You don't even need to pull it out; you get damaged for entering a space adjacent to an enemy with Aura even if you were previously adjacent to that enemy.

This still only gets you 1 damage per fatigue, which is only slightly better than adding more black dice to an attack.