.... But that's not an alliance rule and it only applies to their elites.

Anyway, when (I'm tempted not to say 'if') you are turned away from the lounge, make it clear that BA Silver is OW Sapphire.

As a silver travelling in CW, I was turned away at KUL because MH's silver level is OW Ruby and CW didn't contain the magic word 'business.' It got fixed but you will probably need to insist politely, and focus on your Sapphire, not Silver status even though they are functionally the same thing.