Erin,
No, the Disney Gift Card is separate from your Key to the World card, and transferring the credit cannot be done. What you can do is charge your meals to your room key, and then go down to the resort desk with your Gift Card and use it there to settle the bill during your visit.
IMPORTANT NOTE! Don't wait until your last night, as they often run your bill for your stay the night before you leave. They may have so many guests leaving on that next day, and they have to be prepared to stick the bills under the doors, that they start running them earlier in the day, just to get them done in time. If you wait until then, it is too late.
But if you go down a few days before your departure, you can pay off some of those incidental charges on your resort bill with the card then, and that should take care of the bulk of your problem.
I hope that this helps? Not ideal, but not as much of a hassle as you might think.
Have a magical vacation!