[Backport 2.x] [Extensions] Fix timeout error when adding a document to an index with extension running

[opensearch-trigger-bot]

Backport 0eee9a9 from #6275.

[github-actions]

Gradle Check (Jenkins) Run Completed with:

• RESULT: UNSTABLE ❕
• TEST FAILURES:

      1 org.opensearch.smoketest.SmokeTestMultiNodeClientYamlTestSuiteIT.test {yaml=pit/10_basic/Delete all}

• URL: https://build.ci.opensearch.org/job/gradle-check/11097/
• CommitID: b308cdb
  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-commenter]

Codecov Report

Merging #6307 (b308cdb) into 2.x (ce12cb5) will increase coverage by 0.06%.
The diff coverage is 0.00%.

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@@             Coverage Diff              @@
##                2.x    #6307      +/-   ##
============================================
+ Coverage     70.45%   70.51%   +0.06%     
- Complexity    59248    59269      +21     
============================================
  Files          4796     4796              
  Lines        284095   284096       +1     
  Branches      41310    41311       +1     
============================================
+ Hits         200168   200344     +176     
+ Misses        67233    67025     -208     
- Partials      16694    16727      +33     

Impacted Files	Coverage Δ
...a/org/opensearch/extensions/ExtensionsManager.java	47.08% <0.00%> (-0.18%)	⬇️
.../java/org/opensearch/node/NodeClosedException.java	50.00% <0.00%> (-50.00%)	⬇️
...search/aggregations/pipeline/HoltWintersModel.java	21.47% <0.00%> (-34.90%)	⬇️
...nsearch/index/shard/IndexShardClosedException.java	66.66% <0.00%> (-33.34%)	⬇️
...min/cluster/snapshots/get/GetSnapshotsRequest.java	52.63% <0.00%> (-31.58%)	⬇️
...opensearch/snapshots/SnapshotMissingException.java	28.57% <0.00%> (-28.58%)	⬇️
...rc/main/java/org/opensearch/ingest/IngestInfo.java	51.72% <0.00%> (-27.59%)	⬇️
...earch/client/indices/GetIndexTemplatesRequest.java	50.00% <0.00%> (-26.93%)	⬇️
...opensearch/snapshots/SnapshotRestoreException.java	25.00% <0.00%> (-25.00%)	⬇️
...iscovery/ClusterManagerNotDiscoveredException.java	22.22% <0.00%> (-22.23%)	⬇️
... and 475 more

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