As an aromatherapist I would not use any kind of mint on a small child at all. It's not a very skin friendly oil.
Diluted tea tree (such as diluted in shampoo) is normally ok and it is a good oil for both treatment and prevention of head lice. But there are people who do get a skin sensitivity to tea tree, especially if it is applied neat. You could try changing to a different brand of tea tree shampoo to see if that helps. Sometimes it can depend on the environmental conditions in which the trees were grown.
Are you treating an actual infestation or just trying to prevent a recurrence? There are much nicer and gentler essential oils that you can use in either case e.g. bergamot, lavender, and geranium.
For general prevention, there is a nice organic brand of shampoo you can get at health food shops called Faith in Nature and they do a lovely Lavender and Geranium shampoo plus a conditioner. Being organic, they have very low chemicals and they have a website. Or you can make up your own final rinse water just add 5 drops of your chosen oil or oils to 300ml (just over half a pint) of water and use as a final rinse (i.e. leave in hair). Also, put in an atomiser / pump spray and shake really well before applying to hair, paying special attention to back of ears and nape of neck areas.
For treatment, put 10 drops of essential oil(s) into a tablespoon of conditioner, rub in well, pop on a shower cap or clingfilm (or shopping bag!), or just leave your child playing in the bath, for at least half an hour then rinse out and you should see lots of dead bodies in the tub!
Tisserand or Penny Price Aromatherapy sell high purity essential oils at good prices. Both have websites.