Fix publish cancellation handling in MQTT

[bdraco]

Proposed change

backport of https://github.com/home-assistant/core/pull/120826 directly to rc since the test file has changed location.

If publish was cancelled, cancellation should not be propagated to the mid callback. fixes #120805

Alternative is tagging #120826 one for 2024.7.0 but tag and include the test changes below as well:

• https://github.com/home-assistant/core/pull/120574
• https://github.com/home-assistant/core/pull/120636

Type of change

• [ ] Dependency upgrade
• [x] Bugfix (non-breaking change which fixes an issue)
• [ ] New integration (thank you!)
• [ ] New feature (which adds functionality to an existing integration)
• [ ] Deprecation (breaking change to happen in the future)
• [ ] 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:

Checklist

• [x] The code change is tested and works locally.
• [ ] Local tests pass. Your PR cannot be merged unless tests pass
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• [x] Tests have been added to verify that the new code works.

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• [ ] Documentation added/updated for www.home-assistant.io

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