My grandson had constipation issues and the doctor told us that cheese is constipating. So are bananas. There are several foods his mother was told to not feed him. Google "constipating foods" on the Internet. She also said to limit his milk intake to 24 oz/day. My grandson loved milk and cheese and limiting these for him was difficult. I don't know that anyone caring for him were able to do that He still has difficulty with constipation from time to time. He's 7.
Add as much fiber as possible to his diet. Flax seeds added to his cereal or anything in which they'll blend. is a good idea. Psyllium seed, brand name is Metamucil, helps. It does thicken whatever you put it in. You could try putting it in juice and having him eat it like a pudding.
Then there is ...........I cannot think of it's name right now. It used to be prescribed but now it's OTC. It works by absorbing moisture into the bowel which then loosens the stools. It's not a laxative which is of concern to many who think of it as a laxative. You stir it into to liquid and mushy foods. It has no taste, does not thicken the drink or food.
Rice milk is a good alternative to milk. My granddaughter was allergic to the protein in milk and drank rice milk until she was 5 or 6. She still likes it. It's fortified and so has all of the nutrients that cow's milk has.
Are you sure he's lactose intolerant and not allergic to milk protein? It does take a test to determine milk protein allergy. Milk protein allergies are frequently outgrown but not usually by a 1 yo. I'm lactose intolerant and I don't remember ever hearing that babies outgrow it. It's caused by a lack of enzymes in our digestive track. I've tried drinking lactose free milk and still get cramps, gas and diarrhea with it. I can eat cheese and yogurt without getting symptoms.
My granddaughter who was allergic to milk protein could eat cheese when milk caused cramps, diarrhea and gas. Usually, lactose intolerance causes diarrhea. That is why I wonder if he's allergic instead of intolerant.
Or lactose intolerance or milk protein allergy doesn't have anything to do with his constipation. They didn't for my grandson. He was limited in his milk intake because milk can cause constipation.
I suggest loading him up every day with fiber and liquids. Prunes are potent. Try buying dried prunes and cooking them in water or just leaving them sit overnight in water. Once rehydrated they taste good. They're sweeter than the juice. Apparently having just a bit hasn't helped. And, it might take longer than a week to see a difference. I wish I could remember the name of the stuff that is colorless, has no taste and is easily stirred into drink and food. It helped a lot. Other moms have mentioned it on this site.