Re-reading your post and mine, it looks like I misunderstood the direction of travel that you were asking about for baggage.
As far as AV interlining to BA is concerned, if AV are prepared to through-check the bag then it's a transfer which ought to take place. So I'm not sure why BA should be kicking up a fuss about this at MAN.
AFAIK, it's not the passenger's responsibility to prevent AV from through-checking a bag on an itinerary that AV is prepared to through-check it - although I'll be happily corrected if I'm wrong about that.
The only real possibility I can think of that would explain BA's is that BA and AV don't have an interline baggage agreement at all, so that there should be no interlining at all between BA and AV. But in that case, that would apply whether or not you're flying on a through ticket, and has nothing to do with the fact that you're buying two separate tickets.
Having gained access to some more tools to look at the trips, why don't you see if you can fly IB to MAD and then BA to LON? You could even do the MAD-LON sector on an IB code if you book it on a BA-operated service. That way, you'd have access to the full range of IB fares, not just the expensive BA fares that are interline-combinable. That might work out cheaper for you.
Alternatively, if this is practicable for you, it looks like there's a range of US carriers that would connect online via US ports.