You could try calling this doctor and ask if he/she has patients with BCBS who get reimbursed. My LLMD is an internist and does not accept insurance, but I have BCBS and the reimburse me at 70% and then once I meet my deductible, 100%.
I'm sure the doctor will understand that money is an issue and may be able to offer some insight into the matter - of course there's no 100% guarantee about out-of-network insurance, but it might be a place to start!
Or there's always the gray area option of starting with this doctor if you can afford to do so, and continue looking for one who does take insurance, though, I will say that is a tough thing to find, unfortunately. Good luck! I know these decisions are never, ever easy!