Rebecca,
Officially, no, your child would have to pick from the child menus, not the adult ones.
That is the "bad news".
The "good news" is that the table service restaurants at Disney, for the most part, really set the bar for great service. I know of plenty of occasions where we explained specific nutritional needs and they either allowed EXACTLY what you have suggested here, or found even better alternatives.
I'll guess you'll be successful with this about 80% of the time. I suggest;
- Never expect that you will get this, as it is "officially" against plan rules
- Be very clear in your explanation of why you are asking for the switch
- Give the cast member the chance to wow you with great service
I think you will be pleased with the results!
A final suggestion. Before you leave on your trip, call the first high-end table service restaurant on your itinerary. Ask them if they can assist with your daughter's health needs. They will often not only make sure that the menu that day is personalized to her needs, but will often contact other restaurants on your schedule to pass word along.
Not a guarantee, but I have known it to happen often!
Have a magical (and delicious) vacation!