There appears to be some element of discretion here. BA's codeshares (and some other services) should still be bookable via the MultiCity tool, but I'd be the first to say it is painful. If in theory it could be done via MultiCity but doesn't pick up the ITA outcome then I think that's a good reason for asking for the charge not to appear.
The other option would be to use the ticketing desks land or airside in LHR, or landside in LGW. These technically have a £15 booking fee but the agent may waive this if you're nice.