You don't say what your dates are, but if I pick my mid-week in March I find Business seats for the whole trip (LAX-BOS-LHR-DXB) including BA's new ClubSuite on LHR-DXB.
It does seem as though AA is not loading BA inventory into its system (except for BOS-LHR) - if you use baredemptionfinder.com you will see that there are seats available on LAX-LHR (on BA flights, with surcharges). On March 18-24 you can do LAX-LHR-DXB in Club World on all flights for 225000 Avios and $1500 surcharges – how you get aa.com (or a phone agent) to offer you that with AA miles, I have no idea, I'm afraid.