Booking transatlantic with different classes
I'm trying to book a multicity flight (ZRH->NYC, NYC->LHR, LCY->ZRH) booking the ZRH->NYC portion in World Traveller Plus (well, it's changing at LHR and don't care if ZRH->LHR is in Economy), and the NYC->LHR portion in business. I cannot persuade the BA web site to do this (indeed for some reason it insists on a WTP booking without mixed classes that the LCY->ZRH segment is routed via Dublin, which is pretty strange). I can't persuade any other travel search engine to do this (tried at least SkyScanner, Opodo, Kayak). Kayak will find me the fares, but only via AA, which would be fine if it weren't for the fact that the AA website says "no multiclass bookings". Even booking the both transatlantic segments in WTP seems fantastically hard (I can't get the BA website to give me a direct LCY->ZRH flight at all).
Is there some magic interface to the BA web site where you can just tell it what flights and classes of travel you want? Sometimes I can get to something like this from Kayak, but it then insists I have another BA booking running in another window (I don't) and refuses to proceed.
Alternatively, is there a "I know what I am doing" booking website via an agent?