tests(clustering): sync v1 should work with or without privileged agent

[chronolaw]

Summary

Refactored some code in #13857

KAG-5807
KAG-5811

Checklist

• The Pull Request has tests
• A changelog file has been created under changelog/unreleased/kong or skip-changelog label added on PR if changelog is unnecessary. README.md
• There is a user-facing docs PR against https://github.com/Kong/docs.konghq.com - PUT DOCS PR HERE

Issue reference

Fix #[issue number]

[team-gateway-bot]

Cherry-pick failed for master, because it was unable to cherry-pick the commit(s).

Please cherry-pick the changes locally.

git remote add upstream https://github.com/kong/kong-ee
git fetch upstream master
git worktree add -d .worktree/cherry-pick-13866-to-master-to-upstream upstream/master
cd .worktree/cherry-pick-13866-to-master-to-upstream
git checkout -b cherry-pick-13866-to-master-to-upstream
ancref=$(git merge-base f494c53f8b0d8c9991c8c99294251feb5e164e45 e0cd1b361586e852d19c9dad631ba8e78e0f9122)
git cherry-pick -x $ancref..e0cd1b361586e852d19c9dad631ba8e78e0f9122

[chronolaw]

Wait for #13857 cherry-pick successfully.

[team-gateway-bot]

Successfully created cherry-pick PR for master:

• https://github.com/kong/kong-ee/pull/10761