You will need to include your BAEC in the booking to access advance seat allocation and your other Sapphire perks. There has been discussion on these forums suggesting that oneworld carriers allow status recognition of one program while crediting flights to another, however I have never attempted this and there is the risk of flights being incorrectly credited.

Whilst Dave Noble has rightly highlighted the considerable benefits of joining the AAdvantage, something else to consider is when flying domestically in the US is that AA status passengers are not granted lounge access, however QF or BA status passengers on AA domestic flights are. I only mention this as you noted a trip to ORD, which would most likely involve AA domestic sectors. So on these flights if you were AA Platinum you wouldn't get lounge access, whereas as BA Silver you would.