The BA program will be a problem if you are unlikely to make the four aualifying BA flight (min fare type) in the following year's to be able to retain status. But if you are likley to fly a DONE4 each year, then that should not be too much of a problem so long as you route accordingly. But looking at your itinerary, there are possibly only two BA flights (assuming using Aer Lingus for DUB-LON-DUB).

Do you have any OneWorld status now? If not, the taking the AA Plat Challenge will get you lounge access for the Australian domestic sectors by the time you down here. Lounge access is not granted to business class passengers on domestic flights. But it is granted to OneWorld Saphire and Emerald members. And the Qantas Pubs are worthwhile visiting. It might seem strange since there are no AA flights on your itinerary!

Note that most of the OneWorld (perhaps all?) programs require a minimum of 4 sectors per year on your selected airline to retain status. BA seems to be the most stringent about applying that rule. I have not heard of others enforcing it, but it is generally written into their rules. And BA requires that those 4 flights be at a minimum fare level, so the cheap discount flights don't count (nor do they earn miles or status). They don't let you move up to the next status tier until you have completed the four BA qualifying flights, and you may lose a lot of status credits along the way while you wait since they reset back to zero from whatever level you have when you evenutally move up to the next tier.

So BA is good if you will be flying in premium classes (such as your DONE4) and can ensure you meet the 4 BA sector rule. AA is good to get OneWorld status quickly (via challenge). I don't think you have enough flights before you hit Australia to make AQF Gold (OneWorld Saphire), so again Qantas Club lounge access won't be available on domestic flights)