That ship sailed decades ago. Too much software and middleware expects GET to not have a body and who knows how itll break when you start sending one. Obviously you can do it today and it might work and then randomly break when the code between client and server changes.

Adding a new http method is the only way to support something like this safely. If something in between doesn't know what to do with QUERY it can just respond with a 501.

Fun fact - GET and HEAD are the only required methods one needs to implement to be an http server. It is a pretty low bar :)