Hi, I'm still learning my way around but having fun. I noticed that during a battle I can click my Inventory regardless of whether my team is battling Sylesties or bandits. Traps do not work on bandits. Is there something I can buy or get, like the avatar handheld items, that will let me bash bandits on the head, or cast spells on them, or anything like that?

Sadly, there is nothing much your avatar can do while in battle. The battle's meant to be fought by your pets. Whether there will be changes to this in the future, I do not know.

Traps will only work on un-named pets.
Avatar items are solely for the looks of the avatar, and confers no extra usage/advantage.
Your pet will have some spells once they reach a higher level. But there are no spells for the avatar to use.
The avatar can only trap pets, scan pets, and heal your own party.