[SDK Generation Pipeline] Support integration testing for the SDK repo changes

The SDK generation pipeline previously supported running integration tests using a custom branch in the Azure SDK repository (not just the forked `azure-sdk` repo), allowing us to validate SDK repo changes directly in the pipeline.

However, after implementing the feature that pushes SDK code changes to an existing branch, the pipeline was changed to support only the forked `azure-sdk` repository.

The `azure-sdk` forked repo currently grants write permissions only to the bot account, which prevents developers from creating branches for integration testing. We need to address this limitation to re-enable integration testing—either by relaxing the permission restrictions in the forked SDK repo or by updating the pipeline to support using a branch in the non-forked SDK repo.