2-3 week is a reasonable estimation for Premium Process H1B tran

You do not need to wait the transferred to be approved to work. So, in your case premium process ($1000 more) is not necessary. But as I know, usually HR staff in universities had poor knowledge on this, and they might be very stubborn and insist their wrong opinions. So, if you have failed to persuade them that you can work after the petition filed, then I would suggest that you insist that they do premium process for you.