I have both accounts also but I am not an expert.

I wanted to book JFK-LAX business or first class and I first looked at the AA website as AA does fly this route but BA does not. I saw availability for first in one direction and business in the other. I also looked at the miles needed if I booked on AA using AA miles.

I then looked at the BA website and they do allow you to search for award flights on partner airlines. They showed the exact same flight availability as AA with similar mileage requirements. Their cost per ticket was also $5, the same as on AA. I booked on the BA website as I have miles on BA that I may not be using for anything else. I did this booking two days ago.

I don't think you can combine the miles you have to book one ticket. You can use AA miles for one direction and BA for the other but I don't think that if you needed 240k miles for one flight you can in any way combine the miles.