My wife sees a lot of doctors. We frequently get bills 9+ months after service. Why? In the past, we had my wife double covered by insurance, both under her own and under mine. So the doctor's office would file with her insurance, get back what that insurance would pay, have to recalculate the remaining balance and then send it to the second insurance. When it finally came back from that one, they would evaluate if there were any overage fees that they would waive and would finally issue us a statement to be paid. And if other of the insurance companies rejected the insurance, they would have to review the reason for rejection and if it wasn't for a valid reason, they would file an appeal for us to try to make sure that we paid as little as possible.
I, for one, appreciated how much the doctors offices tried to save us money by pushing through the bureaucratic red tape associated with medical insurance. I get annoyed if they send us a bill quickly and there's a question about if the insurance deducted the correct amount and the office didn't bother to see if anything could be done, but just forwarded us a bill for several hundred dollars when it should be much less.
I would definitely pay. If you don't pay, they will probably send it to a collection agency and you credit will correctly be dinged.