Well, my son was low iron and constipation, yes. In his case it was actually just delayed gastric emptying (slow transit time). Low liquid intake can cause it as could too low fat or not enough "roughage" type foods. Have you kept a food diary?
It's not allergies or intolerance in this child. His twin does have allergy and intolerance issues but never struggled with constipation so I can't help there. Does your child show other allergy signs? That would lean me more toward investigating food allergies than just constipation though I suppose it's possible that it's the only sign in someone. It is curious that it strated when you were probably introducing more solids?
Another thought is to do a comprehensive stool test and see if you get a hit on something that might be causing the issue.
The low iron--is he taking supplements? Most of them are constipating. Ferrous bis-glycinate (ferrochel) is very good at raising levels and easy on the GI system (not constipating). It's all I'll use with my child. The other forms did contribute to the constipation.
As far as iron deficiency. I am glad they did celiac test (that would explain both of those). I think it's worth looking for causes if you are certain he is getting enough iron in the diet? Is he? Eggs, nuts, greens like spinach, and seeds all inhibit iron absorption even though they contain iron themselves. When you're looking at diet sources count supplement and iron fortified foods (like cereal with iron added) and the iron content of meat. He may be not getting enough intake to keep up with growth.