SECOND EDIT (in response to you edit)
To format the expression differently, simply modify the string after the :argument keyword in your magit:--grep-expression argument definition (e.g. make it "Expression: ")
To add the prefix to the magit-dispatch popup, simply do it like I explained in the original answer (replace counsel-projectile-rg by magit-grep)
EDIT
To bind a command directly in the magit-status-mode-map, bind it as follows using define-key:
(define-key magit-status-mode-map "G" 'counsel-projectile-rg) I have bound it under G as g is already used as a prefix command in the magit-status buffer.
END EDIT
You can add a transient suffix with the following line of code:
(transient-insert-suffix 'magit-dispatch (kbd "h") '("G" "grep" counsel-projectile-rg)) read the docstring of transient-insert-suffix for an explanation. There is also the command transient-append-suffix which works similarly.
Here I am adding the command counsel-projectile-rg, which I use myself for this case (well actually I am just using the Spacemacs key-sequence SPC /), and which I can really recommend. There are equally good alternatives for helm, consult and probably more... also for other backends like grep, ag etc...