Would it be possible to release my copyright (essentially put it in the public domain) but still have the software licensed under the GNU GPL?
Extract from licence (gpl3):
“All rights granted under this License are granted for the term of copyright on the Program, ”
“Developers that use the GNU GPL protect your rights with two steps: (1) assert copyright on the software, and (2) offer you this License giving you legal permission to copy, distribute and/or modify it.”
From the above it is clear, that you need to hold copyright.
No. As per dannii's comment, somebody (or some legal entity "somebody") has to hold the copyright. Check out "copyright basics" https://www.copyright.gov/circs/circ01.pdf and copyrights for computer programs https://www.copyright.gov/circs/circ61.pdf and Form-TX https://www.copyright.gov/forms/formtx.pdf which is what you have to submit for copyright registration. It's simply not possible to put the copyright itself in public domain.
And, by the way, you should >>always<< register the copyright. In principle, you automatically own the copyright to anything you write the moment it's on paper. In practice, should the matter ever come under dispute, the courts look >>very<< favorably on you for demonstrating your intent by registering the copyright before infringement occurs.