Make it easier to integration test config options

Our integration tests all run with a single config, which makes it hard to integration test a change like #252. Ideally we should do some reworking so we can test them against several configs, like we do in the snapshot tests. We'll have to think about what the best way is.

Once we do, we should add integration tests for #252, which should be fairly easy once the tooling is there.

Compare how integration tests configure and run against the single current configuration with the snapshot tests that already support several configurations. Decide how the integration test tooling should support multiple configs, then add coverage for the change described in #252 and verify the integration suite runs against each configuration.

Written by the indexing model from the issue text.