Use a containerized test suite to test more backends

Currently, Postgres is the only backend that is tested in the test suite for this plugin. While this plugin should theoretically work with any database that is supported by SQLAlchemy, it would be nice to have verification that this is true, both to be able to officially support multiple backends and to protect against future changes that might break support for other common databases.

Containerizing the test suite would allow us to test against multiple database backends. At minimum, we should support Postgres, SQLite, and MySQL.

Review the existing test suite and its current Postgres setup first; the issue does not name specific files or test entry points. Identify how the suite can run against containerized Postgres, SQLite, and MySQL. Done means the supported backends are tested and future changes are protected by those tests.

Written by the indexing model from the issue text.