M.S.
My dd will be 3 in right before Halloween and she has this problem is she drinks milk. She loves it though, so we switched her to goat and she is not having any problems. I don't know about the lactose free milk though, it is certainly cheaper than goat. I don't know if you get WIC or not, but they will switch to goat no problem, but lactose free you need a prescription from the doctor. Also, we have been using Miralax off and on for dd, but she is older. I don't know about a 12 month old. It can't hurt to ask.