Sorry, I doubt that I can be of much real help here.
I would suggest going into the code of copy-url and seeing what mechanism is used to get the text, then writing your own function that gets the text using that method and piping it to xclip or something.
I've done something quite similar to that with xsel recently, and it worked pretty well as a means to get text from a mix of both emacs buffers and exwm buffers. From what I saw in the Next documentation it should be possible to do the same.
I just used an async-shell-command elisp function, and I'm sure Common Lisp would have an equivalent that probably works better, too.