[PR #10933/597161d backport][3.12] Fix flakey client functional keep alive tests

[bdraco]

(cherry picked from commit 597161d)

[codspeed-hq]

CodSpeed Performance Report

Merging #10937 will not alter performance

_{Comparing patchback/backports/3.12/597161d1127d04e7842642d7548d4bd8de3115a6/pr-10933 (cfda266) with 3.12 (0c16102)}

Summary

✅ 59 untouched benchmarks

[codecov]

Codecov Report

All modified and coverable lines are covered by tests ✅

Project coverage is 98.16%. Comparing base (0c16102) to head (cfda266).
Report is 2 commits behind head on 3.12.

✅ All tests successful. No failed tests found.

Additional details and impacted files

@@           Coverage Diff           @@
##             3.12   #10937   +/-   ##
=======================================
  Coverage   98.16%   98.16%           
=======================================
  Files         130      130           
  Lines       39652    39652           
  Branches     2240     2240           
=======================================
  Hits        38925    38925           
  Misses        555      555           
  Partials      172      172           

Flag	Coverage Δ
CI-GHA	98.05% <100.00%> (ø)
OS-Linux	97.78% <100.00%> (ø)
OS-Windows	95.26% <100.00%> (ø)
OS-macOS	96.95% <100.00%> (ø)
Py-3.10.11	96.84% <100.00%> (+<0.01%)	⬆️
Py-3.10.17	97.37% <100.00%> (-0.01%)	⬇️
Py-3.11.12	97.48% <57.14%> (+0.04%)	⬆️
Py-3.11.9	96.93% <57.14%> (-0.01%)	⬇️
Py-3.12.10	97.84% <100.00%> (ø)
Py-3.13.3	97.83% <100.00%> (ø)
Py-3.9.13	96.72% <100.00%> (+<0.01%)	⬆️
Py-3.9.22	97.25% <100.00%> (ø)
Py-pypy7.3.16	82.29% <100.00%> (-6.35%)	⬇️
VM-macos	96.95% <100.00%> (ø)
VM-ubuntu	97.78% <100.00%> (ø)
VM-windows	95.26% <100.00%> (ø)

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