The AA-BA mashup is tricky, yes, but not as devious as it may seem. Between New York and Paris, for instance, AA has only two non-stops a day; BA has a half-dozen (or more, including Open Skies). And, BA has many seats LON-PAR, so availability is greater.
A similar situation exists on this side of the ocean, where BA feeds many AA domestic flights. It's sometimes possible to get around better if you don't mind one stop...we've used AA to Madrid paired with Iberia to Paris (Iberia's charges are much lower than BA's...don't know why), and AA to Barcelona and then a cheap Vueling flight to Marseille. All to avoid BA.
For our trip next spring to England, I had to play around with the days quite a bit to get AA on both the NYC-MAN and LON-NYC legs; not using BA saved about $1300, but seats were scarce, because the non-stop flights on AA were few.