This is very common. Look at this from a 3 year olds perspective. The toilet is scary. It is huge and not comfortable when you are a small child. It feels like you are going to fall in.
Put the seat up and try to go to the potty this way moms .. This is what it feels like.. Even with a childs potty seat, notice the childs feet dangle. To make a BM most children really need to push using their feet. Try to find a step stool he can place his feet on while on the potty.. He probably really needs this since he has a constipation problem. Also make sure he realizes that if you have a seat insert, it will not pop off while he is on it.
Some children do not like potty chairs because they are afraid of their BMs and the smell. They are used to comments being made about, "wow, this was a full diaper." "Yikes, this is a mess." "Ugh this is really stinky." "Don't touch the dirty diaper it is nasty and full of germs."
Next thing he knows we are asking him to go potty where the BM is close to their body AND expect them to clean themselves. This was a "grown up job" yesterday now we want them to handle it. What are we going to ask them to do next? Drive the car?
I suggest you let him practice flushing the potty so he can get used to the tremendous sound. Remind him to never flush anything but his waste and toilet tissue. Let him know that it does make a lot of sound but he will not ever be able to be flushed away since the pipes are way too small.
Ask what makes him afraid of the potty and really listen. See if the 2 of you can find some solutions.
I am sending you strength and patience. He will get there, it may be on his own time, so be ready.