[Backport 2.x] Change extension integration testing to reflect new initialization sequence (#8145)

[DarshitChanpura]

Manually Backports #8145

Check List

• New functionality includes testing.
  • All tests pass
• New functionality has been documented.
  • New functionality has javadoc added
• Commits are signed per the DCO using --signoff
• Commit changes are listed out in CHANGELOG.md file (See: Changelog)

By submitting this pull request, I confirm that my contribution is made under the terms of the Apache 2.0 license.
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[github-actions]

Gradle Check (Jenkins) Run Completed with:

• RESULT: FAILURE ❌
• URL: https://build.ci.opensearch.org/job/gradle-check/19667/
• CommitID: 40d7b8f
  Please examine the workflow log, locate, and copy-paste the failure(s) below, then iterate to green.
  Is the failure a flaky test unrelated to your change?

[github-actions]

Gradle Check (Jenkins) Run Completed with:

• RESULT: FAILURE ❌
• URL: https://build.ci.opensearch.org/job/gradle-check/19668/
• CommitID: f266f93
  Please examine the workflow log, locate, and copy-paste the failure(s) below, then iterate to green.
  Is the failure a flaky test unrelated to your change?

[github-actions]

Gradle Check (Jenkins) Run Completed with:

• RESULT: UNSTABLE ❕
• TEST FAILURES:

      1 org.opensearch.action.admin.cluster.node.tasks.ResourceAwareTasksTests.testBasicTaskResourceTracking

• URL: https://build.ci.opensearch.org/job/gradle-check/19671/
• CommitID: b7b8aac
  Please review all flaky tests that succeeded after retry and create an issue if one does not already exist to track the flaky failure.