Welcome to FT and the OW forum, svellmann!

There actually is a direct service STL-SXM (AA687) with a stop in MIA. So a very nice routing.
Because BNE-CHC doesn't exists and because it is winter (when are you getting there) I would suggest to use the segment for another destination in Australia. My recommendation would be Lord Howe Island. Not a place many tourists go to and one of the most stunning places in Australia. And a good use for RTW tickets as flights to there are expensive.

If you still want to go to NZ, check whether there is a seasonal service to Queenstown from BNE. Otherwise, do not fly to Brisbane but do SYD-CHC-SYD and do the tour with the GF on a separate ticket.

Regarding the FFP. Yes, AAdvantage is most probably a good choice for your flights. Be aware that a LONE4 books into L class. This means you will get less than 100% of your miles when flying on BA, QF etc. In other words, try to book as much as you can using a AA flight number and you will get much more miles.
Also, you will not get any miles on AA when flying BA metal between the US and UK and no BA miles when flying AA metal on that route.
If you join AA (which I would) then consider buying an AAdmirals Club membership for lounge access (or a Qantas Club membership). Might be a good investment.

Edit: Qantas Club memberships can be obtained at a highly discounted rate via the Australian Frequent Flyer site.