Have you checked with your corporate TA to determine whether the connection you prefer is bookable and what it would cost? This is not leg work you need or ought to be doing. It is entirely possibly a glitch in BA's software and if it is, you will have avoided a problem.
If you are going for a shortish trip, just don't check luggage and the rest of the issues go away. If you need to book two tickets, remember that AA will rebook you under its own unique COC provision if it delivers you late to LAX, even though you do not have a connection.
Not worth worrying about which option until you have the various prices in hand from your TA.