hi there!
the most likely culprit would be milk PROTEIN and not lactose, which is milk SUGAR. She could very well have an INTOLERANCE and not a full blown allergy. A large percent of babies/toddlers w/ issues w/ milk protein ALSO have issues with soy.
there are many kinds of 'milk' you can use...
i always seem to 'blame' milk... Has she had a history of infections (chest, sinuses, ears)?? If yes, this can also lead me to believe she has a problem with milk.
try cutting out ALL milk products for a good month and see if she improves. Remember, if it does nothing, you can always add it back. But check everything - bread, baked goods, soups, sauces, snack foods - these can all contain dairy. The only way to know for sure if milk is a culprit is to remove it ALL.
this means even milk as an ingredient 'hidden' in things. google 'hidden milk ingredients' and you'll find the names of what to look for like whey, rhennet, etc.\
if sleep improves, you'll know it was milk.
but what will she drink?
there is rice milk, hemp milk, almond milk, oat milk, potato milk (don't try soy bcse about 60% of those who cannot tolerate milk protein cannot tolerate soy either).
Rice milk is a good sub, but it doesn't contain enough fat or protein for a toddler. I would likely mix it half and half with hemp milk. Hemp is a great source of protein and omega fatty acids, which are essential for little ones.
You should hopefully be able to find milk-free margarine in your local grocery store. Also I know that coconut yogurt and coconut ice cream is available, but it might only be in specialty store, like Whole Foods.
MAke sure to cook foods in a lot of olive oil (or even try some avocado oil) to add some GOOD FATS into the diet while she's not having milk proteins (lactose free is NOT the same thing; lactose is milk sugar and what you're trying here is to avoid milk PROTEIN).