It does sound possible/probable that it's a food allergy situation, though maybe reflux isn't the problem per se.
It's a common term that people hear but I wish to correct you on the use of the term lactose intolerance, so that you'll be purveying the right information to others. Lactose is baby sugar, found only in mammal milks, and babies would not survive to pass on their genes if they could not digest lactose (until very recently).
The common intolerance/sensitivity/allergy in babies, as well as many adults, is to the proteins in foreign animal milks. Whey and casein are the key ingredients you'll want to avoid, while the fat and sugar products, cream and lactose, can be tainted with some trace proteins and also be problematic for those more highly sensitive. Sometimes there are products sold that say "non-dairy," while they're full of milk proteins. The term "dairy" often just refers to lactose. When buying probiotics to take, to help your child's intestines heal, don't settle for non-dairy. Find the ones that say "no milk."
As children approach 5 to 10 years, it's natural for them to lose the enzyme for digesting lactose: lactase enzyme. Losing the enzyme causes gas when consuming milk. It's a natural weaning mechanism. Adults are not meant to be able to digest the baby milk sugar lactose and most around the world cannot and thus must restrict their dairy intake, for comfort and odor's sake. You and I may be more familiar with adults of Northern European descent --- a genetic pool of people who developed persistence of lactase enzyme through millennia of drinking animal milks to survive cold Northern environments.
One more point about lactose: at times when a child's intestines are inflamed from an illness, a food reaction, or antibiotic use, production of lactase enzyme can temporarily decrease. Formula feeding moms may find partial improvement in symptoms when switching to lactose-free formula, but if milk protein allergies are the problem, then only part of the symptoms will reduce, chiefly the gas and some of the diarrhea, but pain, inflammation, malabsorption, rashes, green stools, reflux and irregular bowels can persist. Switching to a formula with no animal milk proteins in it will allow the return of lactase enzyme as well as eliminating all symptoms related to them.
I was once "such a dairy person" as well. It wasn't hard to avoid it though when I found such a night and day difference in my son, and when I COULD SLEEP because my son could sleep if I didn't consume animal milk, and some other foods that bothered him in my diet. After three years of total milk avoidance, I found the taste of milk to be quite gamey when trying to return to it. By that time I had done enough research on it that I didn't want a lot in my diet anyway. I do some cheese now and then but am quite happy and satisfied without much milk in my diet.