Fix race condition on ingester client metric (#5939) #8942

Job	Run time
lint	19m 10s
lint	19m 10s
lint	19m 10s
CodeQL	6m 36s
CodeQL	6m 36s
CodeQL	6m 36s
build	4m 45s
build	4m 45s
build	4m 45s
test	18m 25s
test	18m 25s
test	18m 25s
integration-configs-db	2m 11s
integration-configs-db	2m 11s
integration-configs-db	2m 11s
integration (requires_docker)	5m 42s
integration (requires_docker)	5m 42s
integration (requires_docker)	5m 42s
integration (integration_alertmanager)	3m 49s
integration (integration_alertmanager)	3m 49s
integration (integration_alertmanager)	3m 49s
integration (integration_backward_compatibility)	8m 27s
integration (integration_backward_compatibility)	8m 36s
integration (integration_backward_compatibility)	8m 51s
integration (integration_memberlist)	8m 24s
integration (integration_memberlist)	8m 24s
integration (integration_memberlist)	8m 24s
integration (integration_querier)	12m 20s
integration (integration_querier)	12m 8s
integration (integration_querier)	12m 8s
integration (integration_ruler)	4m 6s
integration (integration_ruler)	4m 6s
integration (integration_ruler)	4m 6s
integration (integration_query_fuzz)	2m 35s
integration (integration_query_fuzz)	2m 35s
integration (integration_query_fuzz)	2m 35s
deploy_website	1m 32s
deploy_website	1m 32s
deploy_website	1m 32s
deploy	0s
deploy	0s
deploy	3m 24s
	4h 57m 39s

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