QR mixed class booking through BA exec

I'm trying to book a mixed class (business/first) QR reward flight.

As far as I can tell, the options for booking QR reward flights are QR, BA, IB, AY.

QR offer mixed business/first for the same points+money as business/business. So far, so good!
BA's fees for the business class booking are £160 lower than QR. Annoyingly, clicking "First" displays the mixed class itinerary but produces an error message - "there's a problem with booking this journey online".
IB's fees are a bit lower than QR's, but I think give up refundability.
AY don't appear to offer multi-segment QR bookings.

Obviously, if possible, I'd like to get the mixed class booking through BA to take advantage of the lower fees. Has anyone had any success with this, either online or on the phone?

(I got my wife to call BA because it's non-trivial for me to phone them from where I am, but the agent said they couldn't book it. Asking her to speculatively HUCA wouldn't make me popular!).

Obviously, if I have to pay £160 extra to "upgrade" one segment to First, then so be it, but it would be nice to avoid that if possible.

Thanks!