I have sensitive skin and eyes and I've had problems with peppermint oil before. If your daughter is indeed not being melodramatic - and the fact that she does it consistently and claims that it hurts and, as you say, is one tough little cookie - it's possible that even the tiniest bit could be hurting her eyes. Natural doesn't necessarily always mean non-allergenic. She could just have very sensitive eyes.
Could you perhaps try something else? As an experiment, perhaps spray her with water instead but don't tell her it's different. If she carries on about that too, it might be more reassuring to know that she's not actually in pain and is just being a bit dramatic about it.