SQL Server Partially Contained Databases and Linked Servers
There does not seem to be an issue for me querying a Linked Server from a partially contained database when logged in as a contained user. I would always recommend using a synonym. I did this on a stand alone SQL 2014 instance, using 4 part naming or the synonym worked fine. When logged in as a contained user you cannot see the Linked Server, but it works fine to call it.

Don't see why it would be a problem with Always On Availability Groups...just create the Linked server on each replica, and call it. Not sure what you mean by secondary replica, do you just mean in a 2-node AG? Give it a try on your instances and see what happens.