You can actually do this on aa.com, although not all economy fares are combinable with all business class fares.

You may also get very different results if you put the trip into aa.com as a simple RT trip, GCM - UK, or if you put it in as a multi-leg trip with all the sectors listed separately. The latter usually produces the best results and gives you the best chance of avoiding off-the-wall multi-stop routings if you search with the no-stopover option selected.