These banks can offer 2nd chance checking accounts in Rhode Island and they do not use any ChexSystems services(Latest Updated list of banks). But I can not guarantee that you can apply successful. So I provide so many banks in Rhode Island as alternatives for you.
Webster Bank : Westerly, Cranston, Warwick, Providence, East Greenwich, RI. And the bank offers opportunity checking account.
Providence Branch Address: 51 Silver Spring Street, Providence, RI 02904