[ECO-2242] Add unit tests for WebSockets; fix `processor` startup behavior in e2e tests

[xbtmatt]

Description

Adding unit tests for WebSockets.

This PR is stacked on top of #278

• [x] Added unit tests for WebSockets
• [x] Update the docker e2e tests to wait for migrations to be complete by fetching the last successfully processed version

Testing

E2e tests!

Checklist

• [x] Did you update relevant documentation?
• [x] Did you add tests to cover new code or a fixed issue?
• [x] Did you update the changelog?
• [x] Did you check all checkboxes from the linked Linear task?

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Name	Status	Preview	Comments	Updated (UTC)
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