Local AD user session mapped from trusted remote AD authentication

I have a request to obfuscate a customer's usernames. The customer has an AD which is unmanaged by me (local). All I can really do is proxy to them and get a trust relationship set up.Here is what I want to do:

Login to Horizon client using the remote domain, which is trusted by my local domain. The user enters the remote domain credentials, and is then mapped to the local domain as a local (mapped) user and his desktop comes up in his local user domain session.

So user@ㅤ .org logs in on the local.org domain, and the proxy sends the auth request to the trusted remote.org for auth. Authentication is done, and then the local.org domain AD creates a session as the local mapped user - user@rayn .org.

Is this possible at all?