Add E2E and acceptance tests for settings

[justlevine]

What problem does this address?

Gutenberg likes to break things, but we need to make sure updates don't break any functionality or extensibility.

What is your proposed solution?

We should implement codecept acceptance tests and playright e2e tests to ensure a stable backend.

What alternatives have you considered?

No response

Additional Context

Inspiration: https://github.com/WordPress/gutenberg/issues/38851