Use azpysdk Breaking Check in CI

[JennyPng]

#42601

pipelines (i didn't find a mgmt package that doesn't opt out of breaking, lmk if there's one i should check) - these outputs are the same as with tox

• core https://dev.azure.com/azure-sdk/internal/_build/results?buildId=5625635&view=logs&j=b70e5e73-bbb6-5567-0939-8415943fadb9&t=0a38c4fb-2772-5578-aac2-81402a187585 (just opts out of breaking)
• storage https://dev.azure.com/azure-sdk/internal/_build/results?buildId=5672261&view=logs&j=b70e5e73-bbb6-5567-0939-8415943fadb9&t=0a38c4fb-2772-5578-aac2-81402a187585 (opts out)
• ML: https://dev.azure.com/azure-sdk/internal/_build/results?buildId=5684639&view=logs&j=b70e5e73-bbb6-5567-0939-8415943fadb9&t=0a38c4fb-2772-5578-aac2-81402a187585 (opts out)
• azure-ai-contentsafety: https://dev.azure.com/azure-sdk/internal/_build/results?buildId=5675933&view=logs&j=b70e5e73-bbb6-5567-0939-8415943fadb9&t=0a38c4fb-2772-5578-aac2-81402a187585 (other analyze checks are already failing in their other pipelines so not to do with this change...)
• azure-ai-vision-face: https://dev.azure.com/azure-sdk/internal/_build/results?buildId=5675944&view=logs&j=b70e5e73-bbb6-5567-0939-8415943fadb9&t=0a38c4fb-2772-5578-aac2-81402a187585 this just says "No revelant version for azure-ai-vision-face on PyPi. Exiting..." but that's also what Tox outputs

[scbedd]

Lets run an invocation of ml and storage and ensure that it's matching.

[scbedd]

Pull off draft, attempt verify it matches for storage and ml, verify it then I think we're g2g on verification.

[Copilot]

Pull request overview

This pull request migrates the breaking changes CI check from the legacy dispatch_tox.py script to the newer dispatch_checks.py script, which uses the azpysdk breaking check tool instead of tox. This aligns the breaking changes pipeline with the migration effort tracked in issue #42601.

Key Changes

• Switches from tox-based breaking changes detection to azpysdk-based breaking changes detection
• Updates script path and argument names to match the new dispatch framework
• Adds explicit environment variable configuration for proper isolation

[scbedd]

@JennyPng you're done legit as much verification as we can ask for. Any issues that come back will just be fixed in main. Thank you for your patience and all the work on this!

[scbedd]

/check-enforcer override