Ian : There are many different types of interline agreement. I haven’t seen what you’ve seen on the website so you may be right, but in general terms an interline agreement refers to mutual (or sometimes unilateral) acceptance of tickets and other documents.
It would need to specifically say baggage interlining in order to validate your contention.
I would be surprised if they would interline your bags if you are on two separate tickets. I know this is not what you want to hear but they are probably correct in refusing to do so – and I’m not trying to defend BA here, I am no fan of theirs.