Don't know how long you have, but if these are paid premium-class tickets, you might want to think counter-intuitively and head out the other way. Either the NZ-SQ-VS "Escapade" RTW, or the SQ-VS two-carrier RTW products might get you to Paris the fun way at a cheaper price than the AA fare alone. If you can get UA to comp your AA status you could book the SQ miles, and ride Economy+ on UA domestically following the full re-introduction of LRTC on AA. Look at SQ's website here.