Toddler chronic constipation - celiac issue?

Hi everyone. I'm sure we've all seen our fair share of constipation issues with our kids but it seems it's been an everlasting struggle for my daughter.

She's almost 3 and been struggling with constipation since she was about 1.5 years old. It initially got really bad when we introduced cow's milk to her - so upon the recommendation of her paediatrician, we stopping all dairy products, which help alleviate the problem initially. She was fine for about 6-7 months then all of a sudden the constipation issue returned again. Ever since then, she is on the laxative (on and off, not daily). I tried everything - all the recommendations - the fruits, the vegetables, the fiber, the probiotics, the prune juice, etc. - and it seems like the problem is still there.

It's worth mentioning that her dad's side of the family all have celiac disease - gluten issues. It doesn't affect my husband but my mother-in-law and his sisters have celiac disease. Because of this, she is going to do the gluten-test next week upon the recommendation of her GI specialist. In the past couple of days, she has been pointing at her tummy, indicating that she is not comfortable.

Just want to hear from other parents who have been through a similar situation... was your toddler chronic constipation related to celiac disease or gluten intolerance? Is there something else I should be looking out for her? Are your kids on laxative for long periods of time? Does it get better? :( First time mom here, so looking for any support on this issue - thank you!