Has your vet considered that he could be a borderline 'dummy foal'? They can appear that they just don't 'get' it (nursing). If it were my foal I would go ahead and have the vet tube him with a good meal of mama's milk to get him energized. I had a foal like this once and the tubing helped, but I had to keep teaching him to stay and keep nursing, it was difficult because he would just stay at the udder and not suck. It was not fun and time consuming, but the tubing helped to a degree that he perked up a lot. He did finally learn to nurse normally; my daughter and I worked together 'patterning' him by moving him to the mare and offering the teat, milk would just spray all over his face the mare was so full, so frustrating. Your foal is adorable, I hope he will catch on soon.
I know of a lady with this problem, she syringed for a while, and she had to go to giving her foal milk in a bowl which I don't know the difference from nursing because that takes a sucking effort as well I think.
I don't know anything about lactose intolerance, sorry.
One of my routines with my foals was to give them some probiotic at the time the mare came into her foal heat and it made a difference with the diarrhea thing.