First of all, I'm sorry about your daughter's sick Betta. I understand the trauma!
The first thing you need to do if you have not done so already is check the levels of things in your tank. The pH, ammonia, nitrIte, and nitrAte levels all need to be checked. The brand API is a good testing brand. The ammonia and nitrIte levels should be kept at 0 (especially the ammonia). The pH levels for Betta fish should be around 6.8-7.0. If the pH isn't exactly at these levels the fish can adjust . . . a STABLE pH is the main thing. Did you cylce the tank before putting the fish in? If not, then you will see spikes in the amonia and nitrIte levels which will spike causing stress to the fish. After the tank starts to cycle (about 1 month) then those levels will start to stay at 0. Until then do at least a 25% water change when the levels are not at 0. Don't do anymore than a 50% water change as it will stress the fish even more. Another thing that can help stablize the water is Java MOSS which you usually have to purchase online. It basically lives off your fish's poo and helps keep the ammonia and nitrIte levels low. There is a company called Aquarium Garden online where it can be purchased. Also, what is the temp in your tank? The ideal temp for Bettas is around 78 degrees but anywhere in the 70s is fine as long as it is constant.
Since I cannot see the fish it is hard to say what the growing white spot is. Could you describe it a bit more or take a picture? White spots are usually fungal areas but without seeing the fish I can't really say. Along with the Bettafix add Pimafix which is the companion to the Bettafix. The Pimafix will treat any fungal infections naturally. Adding medication to the tank may cure the fish but will also kill off good bacteria that keep your ammonia and nitrite levels low. Going natural in treatment is always better. If you think the fish is constipated, feed him a COOKED, COOLED, and PEELED pea or two. It will act as a laxative for him. It does sound like your fish is ill but don't panic about the eating. Betta fish have been known to go for a month without eating. The Hikari Bio-Gold are the best food for Betta fish and lots of breeders use that. When feeding your fish place the pellets in a small amount of water for about 3 minutes then feed him several at a time. This keeps the food from enlarging inside the fish after he eats it which can cause other problems.
I hope some of this helps and that your fish recovers. By the way, what size is your tank?