Although this sounds odd, find a date on ba.com near when you want to travel where there's 'no availability' then go back search again for that specific date. The site will then offer to 'include partners'. Select that option and then it will show AA availability. From there, search '7 days later' (or earlier of that's where your date is). The results will then show all award availability including partners.

The bad news is that I'm pretty sure you'll still pay the big surcharge if you use Avios. Last time I did that I paid something like 50x more that if I'd used AAdvantage miles for the same trip.