I am not sure your lab child can still free either using UA or using AA program.
I know with AA program the infant fare is 10% of the fare of the cabin the child travels in. I assume you are talking about redemption in coach. It is not too bad. But it can run quite a bit of $ should you redeem a business class award for the adults and then would need to pay 10% on the business class fare for the infant who travels in the business cabin with you... (like over a thousand dollar for the infant alone).
UA should be your best bet especially if you would travel in coach.
There are currently a 50+5K sign up bonus after 1st purchase and adding an authorized user, on Chase UA card that you can sign up. It takes some "work" to be able to see the offer but it is a public offer so once you can see the offer page you are good to go. Check the Miles Buzz forum for its dedicated thread and read both Post 1 and Post 6 for the How To.
Or go to this blog and read about the UA card.
http://millionmilesecrets.com/credit...-credit-cards/
BTW, using AA miles for Australia you would book partner award flying QF - which, contrary to the 330 days myth that being bantering about, if all you need are coach seats, there are plenty of availability. QF flies to SYD/MEL/BNE from LAX. One other alternative is to aim at QF's JFK-LAX-SYD - quite often when LAX-SYD has no award availability there IS availability for JFK-LAX-SYD because of "married segment" that reserves some seats LAX-SYD for redemption from JFK, so folks from East Coast would still have a chance instead of all award seats being taken up by folks on the West Coast.
Your issue though, is getting a Saaver seat from IAD to either LAX or SFO in order to connect to the QF flight - because lately AA has very LIMITED availability even on the domestic Saaver award - for example, between MIA and LAX there are 6 or 7 daily nonstop flights but all I can see have award availability may be 1 or 2 with the rest all require connection(s). You may have a much easier time to get from IAD to JFK on the other hand.
Also please note the above is the CURRENT situation. If your planned travel is 1 to 2 years from now, things may be very different from what they stand as of now. Just keep this in mind.
DL flies to Australia on its own metal as well. And DONT try the 330 days thing as it would NOT work on DL which generally only release its low level award quite close to departure date (think within the same month or 2-3 weeks before).