S.B.
I'm not sure why the doctor made you think that... maybe he was referring to brand name?
For your son, it makes a difference what about the milk is causing him to have tummy troubles. Lactaid, and other lactose-free milks on the market are for those with a lactose intolerance. Meaning they lack the enzymes to break down lactose in their bodies so consuming foods that contain lactose will cause painful bloating and gas. The lactaid milk is regular milk with those enzymes added in so that the lactose is already broken down prior to consumption. If lactose intolerance is not the specific problem then I don't think the Lactaid milk is necessary for your son. If it is, there are other options that are less expensive, including store brands.
I would speak to the pediatrician prior to making any switch and follow their recommendations since they are familiar with your child and his tummy troubles. If all else fails, just take him off milk. He really doesn't need it after 12 months anyhow. Hydrate him with water and get him calcium and vitamin D from other food sources and a multivitamin.