Provider getting the run-around by BCBS

I am an Illinois licensed clinical psychologist and practice in Illinois. I have a patient who is insured by BCBS Michigan and lives in Chicago. I sent my claim in to BCBS Illinois, as I was instructed to do, but they did not pay the claim. The phone system at BCBSIL will not allow me to speak to a customer advocate and I am not getting paid. I spoke with a customer service rep in Michigan and they said that the claim should definitely be sent to BCBSIL. I spent 2 hours trying to get through this terrible web that BCBS has created. The client has established a nice working relationship with me in the past 5 weeks, and I think it would be harmful to terminate just because BCBS is being unresponsive.