you cannot get a side for dessert...anywhere. They don't sub sides for desserts at Disney restaurants (no apps instead of desserts, either, darn the luck.) The only sub I've ever seen allowed at counter service is a bottled water instead of dessert. Now some places have fruit for sale--and will allow fruit as your dessert. Flame Tree does not allow any substitutions on their platters -- you get the cole slaw & the beans, and that's it. You'll have to pay cash if you want fries with your food.
Some places will let you sub sides for sides...like if you don't want fries with your burger, you can sub fruit-my adult daughter was allowed to do that a couple of places, including Cosmic Rays in MK subbed grapes for fries, and POFQ counter service subbed apple slices for fries