I'd go to water-rich foods, which will provide the liquid as well as much-needed fiber. Watermelon and cucumbers are excellent, so are prunes. Berries aren't bad, clementine slices are good - all if you cut them up, of course. If you have to flavor the water with anything, try a little prune juice and not nutrient-free apple juice.
I disagree with applesauce as suggested below. The typical "BRAT" diet for kids with diarrhea is "Bananas, Rice, Applesauce, Toast." Save those for when a kid has an intestinal bug and has really loose stools - the opposite of what you're dealing with. So stay away from applesauce and bananas, as well as cheese, which increase constipation! If you do any rice, do high-fiber brown rice.
Do you ever make oven fries or sweet potato fries? Cut up an Idaho potato, peel and all, either into wedges or into chunks, and toss with a little olive oil or canola oil, then bake in the oven for 20-30 minutes depending on the size of the pieces. You can do this with sweet potatoes too.
When my kid was this age, we were doing finger foods for breakfast. I made high fiber French toast and pancakes. For pancakes, add in berries or cup up prunes (small pieces) and add in something like wheat germ or wheat bran. If you use a mix like Bisquik, add another egg for moisture to compensate for the added fiber. It's delicious! Add in some ground flax seed too - healthy fat plus fiber. Make a bunch, then freeze the extras between layers of waxed paper and just pop them out and reheat as needed. You can put a small amount of real maple syrup on for flavor and no high fructose corn syrup as is in the cheap "pancake syrups." For French toast, use a high fiber bread with a lot of holes, and mix protein powder in the liquid, which gets into the little cavities and boosts the protein content.
If you are not yet into things like chicken fingers, put them on your list - cut fingers or "nugget" sized pieces you cut yourself from larger pieces, dip in beaten egg, and then in a mix of high fiber bread crumb type products - again, wheat germ, bran, flax seed and even finely chopped nuts (almonds, etc.) whenever it's time to introduce them. Quick fry in a healthy oil until browned and crispy on the outside, then finish in the oven. You'd be surprised how much fiber you can add to regular foods to make them healthy.