This is a great question and I'm surprised it hasn't been asked before. Unfortunately I don't have an answer for you, but please let us know if you try doing it and how it goes.
I'm not sure there is an opportunity to have that added. I recently used a gate pass on the other guys at O'Hare and I don't remmber being offered to present my Global Entry card, although I did fly them once last year to Canada so they would have had my KTN if they looked me up from driver's license.
They did however send me through their priority checkpoint. Southwest may be able to have you use the Fly By, if not TSA Pre. (Southwest can allow Fly By, TSA must allow Pre - since you aren't paying TSA fees on a gate pass that seems less likely?) But bring your card and ask anyway, while allowing some extra time.
Home airport MDW, frequent visitor to MCO to see the mouse.