Gluten free mom wondering about my son..

I'm gluten free due to having an intolerance to wheat, I'm being tested for celiac disease right now but anyways, I clearly couldn't eat wheat when I was pregnant with my son who is going to be 5 months on the 9th this month, we've started him on rice cereal and yesterday he started trying bananas, I have such a long list of food allergies that everything worries me because I'm worried I passed it on to him. I've noticed he has been constipated and saw that rice cereal does that and was thinking about oatmeal cereal but since I'm wheat intolerant I was wondering if I should just see if he is okay with it? I also saw that oatmeal cereal can lead to more allergies? Or is there anyways for him to continue with rice cereal but make it so he doesn't get constipated??