You are thinking too much. Whether there is a contractual relationship between A and B is a matter totally between them. In your case, A does not have to enter a subcontractor agreement with B or open a bid for your job. Once B agreed to help and A agreed to pay him, there is a contract formed and it does not have to be in writing. They could have done that for other projects which B would have expectation for A to pay him. That is enough. Bottom line is you did not hire B to do your work. You hired A so your contractual relationship is with A and you have no obligation to B.
As I said, if you did not pay A, he could use it as an excuse, in which case B would file a mechanics lien on your property and try to foreclose it if you do not pay. Since you have paid A in full there is nothing you need to worry about.
My guess is that B wants to get the piece of paper in order to show in court that A is lying and he has no excuse but pay B. It could be a cancelled check or a receipt from A. You can mark off your sensitive information and give to B to help him. If you have paid in full, why should you worry about anybody bringing a groundless lawsuit against you?
