In the Temple of Trials I found two good spears. My final trial was to defeat the Arroyan in unarmed combat, meaning he took my two spears, healing powder, and antidote. Now, outside of Arroyo, I cannot defeat the 'other chosen one' when he is armed with a spear and I have only a flimsey Vault 13 jumpsuit. I have to somehow get my stuff back - where is it?

All the stuff you found in the temple are in chest just outside the room where you fought with the other tribal.