Do you put her on a grown up toilet with a smaller toilet seat on it? Is it a toilet where the poop splashes in the water or do you put her on an little children's potty which you have to empty? I can imagine her being afraid of water splashing back up on her butt. If it is a toilet like that, maybe try putting toilet paper down first in the water and make a 'bed' for the poop.
Maybe it will help to let her in the toilet when you or Alex go on the toilet, you have to get rid of your own shame, but if she sees your example, maybe she will follow.
If all this doesn't work I really think you need to continue the strict route. Maybe put her on the potty as long as it will take for her to go. Don't let her off it before she goes. Do it in the weekend, don't plan anything else and commit to this. Eventually she will have to go. I wish you all the luck. Again it is hard and you are a great mom anyway. This is just something you have to go through.