Unfortunately, you can't use the Asia Miles online booking tool to book AA award flights. Your idea of checking the BA site for availability is good, and then you'll need to call Asia Miles to book.
As for checking how many miles it will take, just enter your origin and destination on the Asia Miles booking tool (
https://www.cathaypacific.com/cx/en_...ts/facade.html) and look at the amount needed for any non-CX flight - it'll be the same amount*.
It's not a matter of you being dumb, but rather that CX IT systems are just questionable sometimes.
*I think there's a third category (multi-carrier awards) for award pricing involving more than two non-CX airlines or something, but I've never had to deal with that and don't know the details. In any case, it won't apply here since you seem to be looking for an AA-only flight.