My son did this same thing! He was almost 3 and 1/2 when he was fully potty trained. He would not go poopy in the little potty or big potty. He was actually afraid to go. And, just like your little one, he was actually in regular underwear - but with him, he would absolutely hold it, not go in his undies and get constipated. And he also asked for a pull up when he had to poop. I got very frustrated, just like you! From what I read some kids think it's actually a part of their bodies and won't do it. Some are just plain afraid of the toilet - or the flushing afterwards. We tried the same things, rewarding, poop chart, taking toys, etc. Nothing worked! But one thing I did know is that I couldn't be upset with him or punish him about it. I guess that's not good at all when they're learning to potty train.
So, here's what we did. We just kept putting his pull up on whenever he asked for it, and we would take him potty to see if he would poop before we put it on. We always, always, made it positive, quit doing everything but the potty chart (he still got stickers for peeing on the potty) and if he didn't have to poop, it was okay. And every time we put him on to go poop, I would say "It's okay to poopy. Pooping is very good and if feels great when you are done!" - something along those lines. One day, he just went! That was it! So no worries, you're little one will go! Just be patient! Until then, hang in there! :) :)