[v4.13] PYTHON-5406 - Use correct client for test (#2378) #2407

Job	Run time
Make SDist	55s
Build wheels for manylinux_x86_64	2m 26s
Build wheels for manylinux_i686	2m 20s
Build wheels for manylinux_aarch64	11m 14s
Build wheels for win_amd6	4m 2s
Build wheels for manylinux_ppc64le	10m 56s
Build wheels for manylinux_s390x	11m 29s
Build wheels for macos	5m 49s
Build wheels for win32	4m 1s
Download Wheels	11s
	53m 23s

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