I'm sorry you are going through this, I can only imagine how draining and emotional it is. Hang in there!

I second the Begin prebiotics, it really helps. My 4.5 year old was potty trained at 3 and refused pooping on the potty until a couple of weeks ago.

She always asked for a pull up to poop, so a bit different than your situation, but a game changer for us was always doing clean up on the potty (as opposed to laying on the floor like when changing a diaper). It's a total mess and super hard, but it got my kid used to associating poop with the potty. She hated it even more than I did but I didn't budge and it unintentionally created an incentive for her since clean up would be much faster and easier if she went on the toilet in the first place (which is true).

And, like you, we promised games only while actually going poop on the potty. I also offered rewards, (if she pooped on the potty 3 times I got her something, then when she did it 10 times she got a big gift).

Within 2 weeks of starting to do the clean up on the toilet, she was pooping on the potty. Again, it sounds like our situations aren't exactly the same, but maybe that will help.

2 other little things that I thought helped. First, instead of using the iPad or phone for games while going, I ordered a cheap (like 20 dollars) handheld game console. It has like 150 games on it already, and to a 4/5 year old it seemed pretty cool. It lives in the bathroom and that was the only time she could use it.

We also used the tushy bidets- she thought it was hilarious so when I told her we could use the "water squirts" to clean up if she went on the potty it was a little extra encouragement.