You are not really changing the license of the original work though. You are not impacting what the author or anyone else can do. You are distributing a copy under different, more restrictive terms.
A bit like me buying a comic book, then offering to sell it to you under the condition that you never let my brother read it and that you make any future owner agree to the same terms. That's perfectly legal, and there is no reason I would need permission from the author (or publisher) to do that
A comic book is a physical object. This analogy doesn’t hold. When you give a comic book to someone, you’re only transferring the copy and its implied license that carries with it. You can set the terms of the physical object, but you can’t change the license of the content within it.
Suppose you get a license to view a copy of a work (streaming or a paid subscription to a newspaper site). Your permission to consume the media begins and ends with the license terms, which allow you to view the work (and, since it’s necessary, to make a transient copy) during the period of the subscription. You don’t get to relicense it to someone else under those terms.
Your redistribution license only applies to the part you own copyright on. Everything else is still under the original license. You can add AGPL but that doesn't take away the BSD license from the parts you didn't create.
I’m not sure it’s so easily severable in this case.