Oh man do I ever understand what you are going through. I have been dealing with this in my now 4¾yo for nearly 2 years. It is beyond frustrating and it doesn't matter how many times I tell myself that getting worked up doesn't help I am still infuriated by it all. And my kid is neurotypical, can't imagine how much processing disorders make it worse. We are past the point of daily medication but not quite at the daily poops. I tried all the reward tactics, none of them worked. In the end, it's been time that has yielded the biggest results, she's gotten older and understands a bit more.
This book helped tremendously. It's not very long, took about an hour to read.
https://smile.amazon.co.uk/dp/B00PY3U0IA/ref=cm_sw_r_cp_apa_9YQoFb7T905HH
What it comes down to is the hard poop over stretches the bowel and fucks with their urge to push, they actually stop getting that sensation and it switches to holding the poop instead. That's what the medication is for, it makes them poop, then over time the bowel shrinks back and the sensation to push returns. Then at the same time you've got to deal with whatever caused the problem in the first place. With my 4yo, it's was a bout of constipation that happened around the time she potty trained. Also dealt with this with the oldest, he was withholding because he was too scared to ask his new teacher to use the toilet.
My little one starts school in 4 weeks and I'm dreading it causing a regression. I might start dosing her now.
You will get through this, one poop at a time.
Potty issues are the worst and I'm going through something similar with my 4 year old as well.
He's been potty trained since around 2 years old. We went on holidays during the last Christmas break and he became a withholding expert.
I don't wanna scare you but you need to get on this asap before things get even worse and you begin to learn words like "Encropresis".
Important note, all the begging, cajoling and/or frustration won't sway a poop withholder. You will be needing the patience of 10 dozen saints to work through this successfully.
Here's what is working for us:
A book on Amazon: https://www.amazon.com/Stool-Withholding-What-Child-Europe-ebook/dp/B00PY3U0IA/
A Facebook group: https://www.facebook.com/groups/209540745755798/
The book helped me understand some of the psychology of withholding kids. And the Facebook group helped with information from many more experienced parents. The group is mainly against a certain medication for constipation. BUT it has a ton of information about alternatives.
With patience and information, this phase will pass.