Someone is misunderstanding something somewhere, and whether it's you or Blue Cross I don't know. Certainly the clinic is misunderstanding; there is nothing whatsoever fraudulent about having two insurance plans.

But there is nothing in the law that says that bills can only be sent to the insurance from the provider.

If you have an actual itemized invoice from the provider, and not just a statement, send it in. Once the mail room opens it, how is the claims adjuster going to know who mailed it in?