If a bank receives a stop payment order on a stolen check, can the bank charge a fee for the stop payment.
Yes, you can charge a fee. However, some banks do not if the customer has issued a replacement check. The point being that if the first check was stolen and it gets paid they know they would have had to fool with it and they are glad to be able to just return it "reported stolen." As noted "payment stopped" sends the wrong message to the presenting bank and its depositor:
The concept of a stop payment suggests that the check in question had originally been issued and authorized by your customer.
If you believe the loss of a single, completed check tips the scales and compromises the account then many (including some of those above) would suggest you simply close the account.