I would offer more water-- increasing fluids is a good first step for dealing with constipation. Not juice, just water. Try letting the LO have a Really Cool Cup, if the LO is resisting drinking more. My DS had this issue, and I wound up tempting him with this enormous insulated coffee mug "just like Daddy's," and he LOVED it, and sucked down so much water.
Try raspberries, blueberries, plums, apricots, grapes, peaches, and other fruits with a lot of fructose. Some kids who won't eat them fresh will eat them if you offer them frozen, so that might be something to try. Oatmeal is full of soluble fiber. Restrict binding foods like bananas or white rice, and cut dairy back to 16 oz a day or less. Also, try increasing the non-processed natural fats-- olive oil, oily fish, grass-fed butter if you can get it, avocado, coconut oil, or any other whole-foods source of dietary fat. Increasing DS's fats and fluids has solved his problem, without us adding or taking away any other foods.