A test using mock objects would not be a very good test.
Why do you say that? What really makes it different from testing with the real object? When DBD::SQLite writes to the database, is that testing Class::DBI or is it testing DBD::SQLite? Class::DBI really should not care what happens after it sends its SQL to the DBD driver, and to properly test Class::DBI, you should check that it generated the SQL you expected it to. By testing a SQLite database you are not really testing Class::DBI, but the side-effects of Class::DBI's interaction with DBD::SQLite.
So why not use something like DBD::Mock? It behaves pretty much like your standard DBD driver, and you can both feed it mock results and read back a history of SQL statements you have run. IMO this would be a much more effective (and less intrusive) test of Class::DBI's functionality. I have actually been using DBD::Mock to test our inhouse OO-Relational mapping tool and I would never go back to using a real DB to test. It has not only increased the quality of my tests, but it takes me less than half the time and code it did to test it with a real database. I recommend it highly.