When your client is on a Virtual Private Network, their access to the internet is across the VPN already. It will access the HTTP service you mention through the VPN, and out through that network's egress point(s).
I presume the site on EC2 behind load balancers is a public site, so the main concern is about search engine indexing. Without knowing what use case you have for this, the simplest is a HTTP 302 redirect in your web application on that path to the DNS name that would resolve over the VPN from the client, but not for anyone else. The HTTP 302status , temporarily moved, will to tell crawlers to not index as it is "temporary," but it may still be indexed as a location eventually. It is unclear according to the documentation I have read.
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