My older daughter did the SAME thing. She would hold it for DAYS... sometimes weeks to avoid going in the potty. I remember taking her to the doctor because she was giving herself bad tummy aches by not going.
I hate to say it, but based on the advice of her pediatrician, what worked for us was the threat of using suppositories to make her go. We only used underwear (no pull-ups or diapers), and she didn't want to feel dirty in her underwear. So, I had to just be firm with her and tell her if she didn't go, we'd have to use the "poopy medicine." Then, I'd just make sure to give her plenty of time in the bathroom... we'd sit in there and read books or whatever while she tried to go.
She still hated it, but eventually, she DID start going, and it wasn't an issue anymore. (admittedly, I DID use the suppositories a time or two with her, so she knew what it involved)
I'm not sure that Kennedy is to that point, so this very well may not be the advice you're looking for.
(and FYI, my daughter was still doing this when she was nearly 4, so I understand the frustration there for sure).
This makes me SO excited to start potty training Chloe. LOL