Your wife could be on a LATAM issued ticket, but LATAM could have issued as a QF flight number (unlikely but worth checking), as then you would have no problem.

In any rate as you and your wife will be checking in with Qantas (as the operating carrier) and as you both will have BPs that will say Qantas (but one most likely will say QFXXX sold as LAXXX in small print), I would happily go to the QF First lounge with a smile and hand over the BPs and say that your wife is your guest. You never know maybe an angel smiles on you that day.

KF