Trigger backup sync when backup is complete

[ludeeus]

Proposed change

When a backup is complete send a websocket message with the backup/sync type to inform the core that the backup was finished and is ready to be synced.

As the core side is not yet merged, the format of the message might change, so keeping this in draft for now.

Type of change

• Dependency upgrade
• Bugfix (non-breaking change which fixes an issue)
• New feature (which adds functionality to the supervisor)
• Breaking change (fix/feature causing existing functionality to break)
• Code quality improvements to existing code or addition of tests

Additional information

• This PR fixes or closes issue: fixes #
• This PR is related to issue:
• Link to documentation pull request:
• Link to cli pull request:
• Link to client library pull request:

Checklist

• The code change is tested and works locally.
• Local tests pass. Your PR cannot be merged unless tests pass
• There is no commented out code in this PR.
• I have followed the development checklist
• The code has been formatted using Ruff (ruff format supervisor tests)
• Tests have been added to verify that the new code works.

If API endpoints or add-on configuration are added/changed:

• Documentation added/updated for developers.home-assistant.io
• CLI updated (if necessary)
• Client library updated (if necessary)