[Extensions] Change extension integration testing to reflect new initialization sequence

[ryanbogan]

Description

Initial changes to ensure integration testing for extensions functions properly, following #8029

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.
For more information on following Developer Certificate of Origin and signing off your commits, please check here.

[github-actions]

Gradle Check (Jenkins) Run Completed with:

• RESULT: ❌
• URL: https://build.ci.opensearch.org/job/gradle-check/17923/
• CommitID: b6c669d
  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.snapshots.DedicatedClusterSnapshotRestoreIT.testIndexDeletionDuringSnapshotCreationInQueue
      1 org.opensearch.remotestore.RemoteStoreRefreshListenerIT.testRemoteRefreshRetryOnFailure
      1 org.opensearch.action.admin.cluster.node.tasks.ResourceAwareTasksTests.testBasicTaskResourceTracking

• URL: https://build.ci.opensearch.org/job/gradle-check/18143/
• CommitID: b6c669d
  Please review all flaky tests that succeeded after retry and create an issue if one does not already exist to track the flaky failure.

[codecov]

Codecov Report

Merging #8145 (b6c669d) into main (b5b0042) will increase coverage by 0.56%.
The diff coverage is 0.00%.

@@             Coverage Diff              @@
##               main    #8145      +/-   ##
============================================
+ Coverage     70.86%   71.42%   +0.56%     
- Complexity    56636    57017     +381     
============================================
  Files          4722     4722              
  Lines        267608   267588      -20     
  Branches      39216    39213       -3     
============================================
+ Hits         189647   191138    +1491     
+ Misses        61922    60635    -1287     
+ Partials      16039    15815     -224     

Impacted Files	Coverage Δ
...nsearch/gradle/testclusters/OpenSearchCluster.java	0.00% <0.00%> (ø)
...opensearch/gradle/testclusters/OpenSearchNode.java	0.00% <0.00%> (ø)
.../gradle/testclusters/TestClusterConfiguration.java	0.00% <ø> (ø)

... and 429 files with indirect coverage changes

[opensearch-trigger-bot]

The backport to 2.x failed:

The process '/usr/bin/git' failed with exit code 128

To backport manually, run these commands in your terminal:

# Fetch latest updates from GitHub
git fetch
# Create a new working tree
git worktree add ../.worktrees/backport-2.x 2.x
# Navigate to the new working tree
pushd ../.worktrees/backport-2.x
# Create a new branch
git switch --create backport/backport-8145-to-2.x
# Cherry-pick the merged commit of this pull request and resolve the conflicts
git cherry-pick -x --mainline 1 52a5e3f6e0ca599e3193807134ea42660ecdd195
# Push it to GitHub
git push --set-upstream origin backport/backport-8145-to-2.x
# Go back to the original working tree
popd
# Delete the working tree
git worktree remove ../.worktrees/backport-2.x

Then, create a pull request where the base branch is 2.x and the compare/head branch is backport/backport-8145-to-2.x.