To answer OP's question, the complaint is always to the operating carrier of the flight which gives rise to the lounge access.

Your contract is with AA. AA has contracts which are facilitated by OW. OW is simply a marketing alliance.

If QF were to wrongfully deny an AA passenger access, the passenger complains to AA and AA takes it up with the carrier or other operator of the lounge in question.

Not only is this the legal solution, but it makes common sense.