I agree with jcrump's approach. Tellers should be trained to look for the date, but supervisors should make the decision to accept or reject the item. The language from the model version of the UCC (see above) allows the drawee bank to pay a stale dated item, but it does not require it to do so. (If the check is "on-us" the decision to accept the item is the equivalent of the decision to pay the check.)
You accept a stale dated check drawn on another bank for cash or deposit with a complete understanding of the fact that the drawee bank has every right to return it to you and your recourse would be limited to the endorser. It's likely that the drawee bank will not even be made aware of the stale date in time to return it by the midnight deadline, but I suggest that fact not be included in your decision making process.
Remember too, you would be entitled to an exception hold on a stale dated check based on "reasonable cause to doubt collectibility." You could also send the item for collection via the mail, but that would virtually assure that the drawee bank would notice the date.
"Oh God, thy sea is so great and my boat is so small." Breton fisherman's prayer in JFK's office.