You can order a side of mashed potatoes in person at Plaza Inn.
Tomato soup, as mentioned. Earl of Sandwich also has soups. Not sure what flavors though. It's usually tomato soup and a weekly special.
The beer cheese soup at Pacific Wharf Cafe is not pureed. It has chunks of bacon, onions, celery, and carrots. They are soft, but there are definitely chunks. The soup also contains flour, although it's not much.
There is currently a potato leek soup at Royal Street Verandah in New Orleans Square. There is also a vegetable gumbo with rice. That one also has veggie chunks, but they are definitely soft.
You can get scrambled eggs on the side at any restaurant that serves them at breakfast time.
You can get yogurts in the resort gift shops.
Jamba Juice in downtown has smoothies. You'd have to probably avoid any that have seeds like raspberries or strawberries, but they can also sub out any fruits you can't have and can also make a custom smoothie. You can boost them with protein powder, but they also have high protein, low fiber options like peanut butter chocolate.
You can get milkshakes and smoothies at Schmoozies in DCA too. Ghirardelli also has milkshakes and frozen hot chocolate.
If you can eat tater tots, then are available at Red Rose Tavern and Earl of Sandwich as a side.
You can also probably have bananas and those are available at all the fruit carts in the parks.