I understand that you believe your customer would not be cooperative in an attempt to have the check replaced with something a bit more current. I think you're correct on that score.
If this is a check on your client's account at B of A, you can deposit it and hope for the best. B of A can pay it, either accidently or deliberately, and it would be a legal posting to the B of A account, all other things being equal.
What could go wrong? The account could be closed or have insufficient funds to pay the check. The customer may have placed a stop order on the check that is still in effect. The bank might actually review the date on the check, see that it's over a year old, and send it on back to your bank for chargeback to your account. But if you don't draw on the funds for a couple weeks (to allow for a normal chargeback interval), you can be assured that it can't come flying back to bite you later, since the bank's payment will have been final.
And, if BofA does bounce it back to you on a timely basis, aside from any chargeback fee that your bank might impose you won't be any worse off than you are now, right?