Elisp 819 bytes

This was a fun challenge. Not quite finished because doesn't use backspace. Most test cases pass. Use this to make your auto-type bot seem more human :)

Ungolfed

#!/usr/bin/emacs --script
(require 'seq)

;; source: https://emacs.stackexchange.com/questions/7148/get-all-regexp-matches-in-buffer-as-a-list
(defun re-seq (regexp string)
  "Get a list of all regexp matches in a string"
  (save-match-data
    (let ((pos 0)
          matches)
      (while (string-match regexp string pos)
        (push (match-string 0 string) matches)
        (setq pos (match-end 0)))
      matches)))

(defun how-quickly-can-you-type-this-string (s)
  ;; get all possible substrings, save right remainder
  (let ((substrings nil))
    (cl-loop for i to (length s) do
             (setq substrings
                   (append
                    (cl-loop for j from i to (1-(length s))
                             when (not (equal j i))
                             collect
                             (list
                              (substring s i j)
                              (substring s j (length s))
                              i)
                             )
                    substrings
                    )
                   )
             )

    ;; count number of matches in right remainder
    (setq substrings
          (cl-loop for ss in substrings collect
                   (cons (length (re-seq (car ss) (cadr ss)))
                         ss
                         )
                   )
          )
    ;; calculate keystrokes
    (princ
     (car
      (sort
       (cl-loop for ss in substrings collect
                (let ((keystrokes (+(cadddr ss)(length(cadr ss)))))
                  ;; ctrl + c if matches exist
                  (if (> (car ss) 0)
                      (setq keystrokes (+ 2 keystrokes))
                    )
                  (while (> (car ss) 0)
                    (let ((match-index (string-match (cadr ss) (caddr ss))))
                      ;; add characters preceding match as keystrokes and 1 for paste
                      (setq keystrokes (+ keystrokes match-index 1))
                      (setf (car ss) (1- (car ss)))
                      (setf (caddr ss) (substring (caddr ss) (+ match-index (length (cadr ss))) (length (caddr ss))))
                      )
                    )
                  ;; add remaining characters as keystrokes
                  (setq keystrokes (+ keystrokes (length (caddr ss))))
                  )
                )
       #'<)
      )
     )
    )
  )

(how-quickly-can-you-type-this-string (pop argv))

Try it online

(require 'seq)(defun r (regexp string)(save-match-data(let ((pos 0)matches)(while (string-match regexp string pos)(push (match-string 0 string) matches)(setq pos (match-end 0)))matches)))(defun q (s)(let ((x nil))(cl-loop for i to (length s) do(setq x(append(cl-loop for j from i to (1-(length s))when (not (equal j i))collect(list(substring s i j)(substring s j (length s))i))x)))(setq x(cl-loop for z in x collect(cons (length (r (car z) (cadr z)))z)))(princ(car(sort(cl-loop for z in x collect(let ((k (+(cadddr z)(length(cadr z)))))(if (> (car z) 0)(setq k (+ 2 k)))(while (> (car z) 0)(let ((m (string-match (cadr z) (caddr z))))(setq k (+ k m 1))(setf (car z) (1- (car z)))(setf (caddr z) (substring (caddr z) (+ m (length (cadr z))) (length (caddr z))))))(setq k (+ k (length (caddr z))))))#'<)))))(q (pop argv))