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I have dealing with the same issue. That is my solution, it is a bit late but it could be useful.

(defun brust-LaTeX-next-error (args)
  (interactive "p")
  (if-let ((active-buffer (TeX-active-buffer)))
      (if (< 15 args)
          (TeX-error-overview)
        (when (< 3 args)
          (save-excursion
            (set-buffer active-buffer)
            (TeX-parse-all-errors)
            (if TeX-error-list
                (message ":::: WARNING :::: There are errors ::::")
              (message ":::: Be happy, your LaTeX code has no errors ::::"))))
        (call-interactively 'TeX-next-error))
    (message "There is no active buffer")))

I guess that AUCTeX creates the list of errors once the compiling precess is finish. So, the problem with latexmk -pvc is that the process never ends and the list is never created.

When you call brust-LaTeX-next-error with prefix argument C-u it creates the list from the active buffer with the same functions that AUCTeX does when the process ends (again I guess). So, they should interact nicely.

As an extra functionality when you call it with double prefix argument C-u C-u a buffer with all errors listed is created.