I think that you have made an OK decison in choosing BMI...BA is an absolute joke...the tier points reward nothing but Premium Class travel. Virgin..well, we won't go there...

If you want to stick with *A, UA might be another good choice...as you will be able to get E+ access as a Gold on all of your TA and TP flights...as well as a good number of system wide upgrades (translation--pay Y, get C) with the amount of traveling you are doing...they fly LAX-LHR (you wouldn't have to deal with LGW) with a connection in IAD or JFK...also fly LAX-Austrailia with a connection in SFO...not too out of the way...something to consider is that they also have an excellent program in the hotels department..also

Good luck with your search!