Does a bank have to accept a stop payment request if the only information the person can provide is the payee because the payee keeps sending debits with varying, random check numbers and varying dollar amounts?

Or can we turn down that stop payment request since we have no reasonable and economical way of being able to ensure we can stop that payment?

Close the account and open a new one.

If you want to review the rules - see your State's version of UCC 4-403