I actually booked with AA despite it coming out more expensive.
I had put the flights I wanted on hold with them and when I called BA, they showed no availability on those very same flights. The flights they could book (and I confirmed via Qantas award search) where much less convenient.

They never mentioned any fees though.

I was able to easily book a QantasLink flight with a call to the BAEC U.S. call center (which I understand is going away soon).

Was not charged an additional phone booking fee (at least as far as I can deduce, as taxes are equal to what ITA shows).

Hope this helps anybody who comes across this in a search (as I did!).

I've seen comparison of Avios bookings to low cost carriers. I like to use Avios for business class. Alice Springs to Sydney, one way, business, is 20K Avios plus $33.25.