I may be doing something wrong, but if you book on the AA.com website, if you put your BA# it clears out all your personal details, including saved forms of payment, so it's like booking for the first time everytime.
If you book using your AA ff# it won't allow you to change it online. If you call to change it or have it done at the airport, make sure your BA# is on the BP. Often times the agents hand it back, and I find they didn't change it correctly. They'll often claim "it's changed in the system" but 100% of the time it ends up crediting to AA. Some agents know what they are doing. If the agent insists it's done and your BA # isn't on the BP, be persistent they should ask another agent. There's always somebody who knows how to do it.
I also find my TSA precheck doesn't come through, so I have to stop at a check-in desk if I want to use the BA # on my BP.
Then good is of course free AC entry as a Silver.
In terms of mileage crediting I find on expensive fares AA is better.
For redeeming, BA seems to me cheaper on shorter first and economy flights than on AA, but hard to compare as earning is not apples to apples and I don't really pay enough attention to know if what II am saying is perception or reality.