linked servers between availability group to a server
You will need to create an identical linked server on each of your 3 Availability Group nodes (server1, server2, server3). Linked servers (like logins, alerts, and jobs) are instance level objects so do not sync between the Availability Group nodes. You need one on each node in case of a failover, and you need to keep them in sync manually if they ever change.

Once you import the data, referencing the linked server in your script on your primary node (Production listener) and preferably with a synonym,  it will sync to the secondary node DBs in the Availability Group automatically. Synonyms are a Best Practice when using linked servers. If your linked server name changes you just update the linked server, not the code that imports the data.