And if you don't comply with the audit you've supposedly agreed to, what's their recourse? They can try to sue for breach of contract... a contract that they can't actually prove you've entered into.
There's pretty much no way the BSA's tactics can actually be legal. It seems to me like they put on a big show and take on a threatening posture, hoping that you'll be convinced to let them gather the evidence they need and then pay them whatever they're asking.
A bit of common sense: can a random individual or company come into your house or place of business and demand to see proof that you're complying with a private contract?
The BSA has no law enforcement powers.