> That can't possibly be a valid claim, right?

It has indeed happened.

https://boingboing.net/2018/09/05/mozart-bach-sorta-mach.htm...

Sony later withdrew their copyright claim.

There are two pieces to copyright when it comes to public domain:

* The work (song) itself -- can't copyright that

* The recording -- you are the copyright owner. No one, without your permission, can re-post your recording

And of course, there is derivative work. You own any portion that is derivative of the original work.