Two comments,
The airlines continue to drop the ball on how they deal with their customers when it comes to code share flights. They have a contract with the passenger, not the operating carrier. It is their terms and conditions that apply to the contract and a reference that "in case of a code share the operating carriers rules apply" leaving it to the passenger to find out for themselves what that actually means, is simply NOT GOOD ENOUGH. (sorry for shouting).
In a different jurisdiction the regulator (US DOT) have warned the airlines that such references are not valid, and many have updated their COCs accordingly. But, obviously, not BA.
It is the airline that chooses to code share, they could interline instead, and when they do they need to make sure the passenger is properly informed of the contract terms that apply to them.
In this particular case BA had the arrogance to say that BDs CoC applies, then refused to honour them.
My second comment is BA is very poor at dealing with service recovery, even with those that have status. I was a Gold Ex club with them when they screwed me about costing me a days holiday and leaving me stranded at Heathrow a day before a CC strike. Not even an apology, they just ignored me until I went away which I have done. I burned all my accumulated miles and have not paid them for a ticket since.
However, don't threaten them that you will never fly with them again, this just gives them a reason to ignore you. Lie to them, why not they've lied to you, and say you would love to fly with them again.