I can see the Catering selection menu just fine at FSDT JFK Gate 703, see the attached screenshot.
As explained so many times on the forum, and of course in the manual, your own airplane code doesn't have any effect on the choice of the ground operator, because it would wrong if you could be followed by your own crew everywhere in the world, so only the airline code of the PARKING has any effect on the ground operator or the catering operator.
You don't say if you changed the airline code in the scenery AFCAD, or if you changed them in the GSX parking customization page but, as already explained in many threads in the forum, if your changes to the codes resulted in only one operator with an higher score, NO MENU will be displayed for it, since it wouldn't make much sense to offer a menu with just one option so, if the highest scoring choice is only one, the menu selection will be skipped so, this is entirely normal.