Yup, you sure can! This is available as the manual send_exception API:
where e is the exception object. 'userCustomData' is of type Dict and can contain whatever data you like including the string of the invalid URL. Full documentation for this is available at https://github.com/MindscapeHQ/raygun4py#sending-functions, and I'm just clarifying this on the official raygun.com docs for you.