R.H.
Well, IMHO, if her hair is falling out already, that raises a red flag to me that the dandruff shampoo getting blamed for kids' hair falling out might be a fallacy. ;)
Regardless, have you tried an elimination diet with her? Excema is a common symptom of food allergy or intolerance i.e. dairy. The trick is that total elimination of a suspected culprit should be employed. There's a lot of hidden dairy out there (lunch meats, bread, medicines, chips, candy, pasta sauce, etc.) and it can take from two to four weeks for consumed dairy protein to clear the system.
Also, dandruff shampoo, as well as pH regulated products for the 'no-tear' feature, can be pretty drying. Aside from just not washing your child's hair everyday, you might have good results trying something for cradle cap, either a special shampoo for it (found in the baby wash area of the store, I've heard Badger Balm is a good one) or plain ol' olive oil and (soak for a few minutes and do some gentle scrubbing). Follow up with a *careful* application of some good, adult shampoo for color-treated hair, and you might see improvement.
HTH & best wishes!