Kudos to you for not bowing down to the "doctor God". A good pediatrician is invaluable in parenting, but a doc that solves every problem with drugs is not the kind you want.
My boys both had constipation problems b/c it's hereditary for them. I chose to make them a fruit drink in the blender every day. I takes about 3 minutes and loads them up with tons of nutrients and fiber - naturally (they can't get too much of any nutrient when it's natural.) I always bought lots of fruit and had lots on hand. I would also add flax oil for the nutrients to help their brain and nervous system as well. My boys responded very well to this. We always saw a difference when I let things slide. Now they are 8 and 11 and it is our habit to have these "smoothies" a couple times a week, at least. I don't know if they out grew their problem, or if it they had some healing from these nutrients throughout their toddler years, but we haven't dealt with the problem in quite some time.
Some examples of drinks would be:
a) berry juice (enough to liquify and blend)
assortment of berries
b) milk, rice milk, soy milk, or coconut juice
mango, strawberries
c) coconut juice
pineapple, mango
d) orange juice
other citrus or strawberries
e) If you have a juicer, a great favorite is fresh apple juice and carrot juice blend. While the juicer drink will provide less fiber that meal, it will make up 20 fold with the nutrients your kid will pack away. (store bought applejuice is not the same flavor, consistancy or value at all.)
There are a hundred wonderful combinations and you won't have to beg him to drink.
Note: Always keep lots of frozen fruits in the freezer. I usually added a banana to any of these b/c I understand it is the common fruit with the most nutrients babies need. We still often use this as a 3rd meal. Sometimes with popcorn (fiber) or toast/peanut butter.
All the best to you. Let me know how things go.