In the exclamation That you be happy! what part of speech is the word that? Is it a conjunction via ellipsis, i.e. "(I wish) that you be happy!" If yes, then does this poke a hole in the idea that subordinating conjunctions necessarily introduce subordinate clauses?
Your phrase is a fragment (not a sentence). It might occur as the answer to a question ("What do you want?"). 'That' is a complementizer -- it makes 'you be happy' the complement of 'what'. This is similar to the demonstrative pronoun function of 'this' or 'that', except that the antecedent is a phrase rather than something in the external environment.