The main risk here is that AA has systems that look for conflicting flights and may decide that since it's impossible for you to simultaneously in the air to fly LAX-DCA and also flying LAX-AUS, one of the tickets is "impossible". This often results in one of the tickets being cancelled. Unfortunately, I think the only real way to avoid this is to have your company change your LAX-AUS return to fly to LAX-DCA. If they're not willing to do this, make them aware that there's a risk that the ticket that they have booked may be cancelled by AA.
Once you have flown your outbound to LAX, this is pretty easy to resolve. Just call AA, have them split them off from the reservation with the rest of your family and cancel your ticket. Then there won't be any conflict with the LAX-DCA flight anymore, and everything from that point on will go normally.