If you have any non-trivial SQL in your cdbi classes, I would test that against a predefined database (SQLite makes creating test databases easy). Any invariants of your non-stock constructors or set_sql queries fall under that.
I don't know what framework you already have for verifying/testing your created HTML, but I would go with something like Test::WWW::Mechanize or Test::HTML::Content (which really lacks functionality I admit) to see that the correct output gets created. This testing of the templates is more an end-to-end test though and not a test of the single snippets, but maybe you can create some stripped-down templates that leave out all the "uninteresting" fluff. Indeed, supplying an alternative configuration/template directory might be enough already for your different tests.