Just called again. Got a very helpful agent who said that the real problem is getting the return TATL to book on an AA code. Their system will do all the other flights but not all AA codeshares are released to BA and visa versa. It seems that this particular flight is designated as BA code only when booking through BA.
Whatever the rights and wrongs are they won't do it and so I'll have to try AA. I don't seem to be able to enter more than 6 sectors on the AA multi-city site.