Unless your wife is in the lobby, you could drop your bags at her hotel at the bell desk and she would be none the wiser.

Other option is check everything and leave them to get collected by Delta left luggage. Probably best to ask if that is a viable option in the DL forum. Your question has nothing to do with CX since they have nothing to do with your problem, you would have the same problem if it was some other airline. The problem has nothing to do with what airline you’re flying after you get to LAX.