Can a kid profile battle another trainer in great league in Pokémon go?

Yes, child profiles are able to do PvP Battle Leagues. If your child's profile is currently unable to use these feature, then you can enable it accordingly through the various portal sites

If your child logs into Pokémon GO using Niantic Kids or Pokémon Trainer Club (PTC), you can opt into enabling social features such as Friends, Gifting, and Trading and Trainer Battles for your child in either the Niantic Kids Parent Portal or Pokémon Trainer Club website. You can also revoke your child’s access to these features at any time.

_{Niantic Support}