@tsullivan @Alex_Piggott @Francisca_Lima and others interested.
Looks like you should use
proxy_set_header Host $proxy_host;
instead of
proxy_set_header Host $host;
$proxy_host takes host value from proxy_pass directive, while $host will contain your server_name. And apparently, Kibana from Elastic Cloud requires an original host to determine target cluster ID.
Works for me without proxifying any additional X-Found-Cluster headers.