I do dummy booking on CX website . It asks to enter last name . Then first name and middle name in one . So if you book on AA , BA or AS for CX flight . It will not take middle name even you have provided middle name . You have to put your middle name after first name and leave middle name blank . Then your full name will be on reservation .
I have a booking with CX using AS miles . I just called to put the middle name after the first name . Now My booking have full name that match with passport.