While being afraid of "losing a part of yourself" may be the case with some kids, I don't think your daughter's issue is as simple as that. We have the exact same issues with our son. He is on Miralax. When you say a small dosage, how much do you mean? We give him a half capful daily, and if he goes two days without a bm then we up it to a half capful twice daily. We started by taking him to the pediatrician who felt his abdomen and ordered an x-ray. We didn't think that constipation was the real issue because he had been passing soft stool infrequently on his own. We were shocked to see that in the x-ray, there was a grapefruit-sized obstruction of stool just above his rectum, and the entire length of his intestines was full of stool waiting to get out. The soft stool that he would pass on his own was the over-processed stool that made its way past the obstruction. We started him on an adult-sized dose of Miralax for about a week and a half, and believe me, it was messy. We are waiting for his rectum to return to its regular size after having been stretched and desensitized to pressure for so, so long. This is a very long process and we are only in the third month of it. It will take much longer.
Diet has a lot to do with this. You're right, cheese and milk are binding. Yogurt is different, though. If she won't eat vegetables, can you sneak them in until she learns to like them more? There are a few juices that contain vegetables as well, like V-8 Fusion and I think Juicy Juice. I also add pureed winter squash to my spaghetti sauce. It makes the sauce thicker and richer, is undetectable and it adds a lot to the meal health-wise. Keep offering vegetables to her at each meal. The saying about needing 10 to 12 exposures to a food is so true. And if she likes fruit, blueberries have a very high amount of fiber. Make sure she gets the right amount of fiber each day - 8 to 10 grams. We love the Yo-Plus yogurt because the probiotics are great for intestinal health, and it also contains 3 grams of fiber per serving. Our son also loves the SunMaid Raisin Bran Muffins. And, make sure your daughter gets plenty of fluids. Prune juice is great! Finally, does the multi-vitamin contain iron? Iron can be binding as well.
My advice is to seek the help of a pediatrician if you haven't already. We've found that constipation to this degree is serious and very difficult to regulate without the help and advice of a doctor.
Let me know if you have any questions! I have a chart of the fiber content of typical foods for a toddler. I'd be happy to send you the info.