Is hau'oli a verb that means "to be happy"? Or is it just that the verb to be is eluded? If so, does 'Olelo hawai'i have a verb to be?
There is no verb to be. The word hau'oli can be used as a noun, adjective, verb, or adverb.
Hauʻoli is a ʻaʻano. ʻAʻano is a adjective, descriptive word, or a stative verb.
Surely You are happy?", and similar contructions, which are common in English, should be acceptable.
The prompt was a question, and Duolingo does not recognize punctuation. So you have to use proper word order in English for questions so that the system knows you are trying to ask a question instead of making a statement.
That proper word order, in context.
"How are you?"
"You are happy? How can you be happy, in a world like this?"
The audio does not sound correct. This is for a question, and the intonation for a yes/no question was not used.
Your own dictionary says "you are happy" for Hau'Oli oe