Thank you for your detailed question! Congratulations to you on how far your son has come with his eye contact, language and interaction. It sounds like you have been doing a wonderful job helping him so far!
It sounds like there are two separate things going on. The first one being control! I have worked with hundreds of children who will hold their poop in. In a world that is unpredictable for him, your son is craving a sense of control and something that he and only he can be in complete control of is when and how he poops, let's face it, no one can force us to poop, it is completely up to us. So to help him with this I would try and do as much one on one playroom time in a non-distracting environment as you can where he gets 99% control in that room. The fact that he is holding his poop in at school also tells me that he is having challenges relaxing in that environment, and it's fantastic that you are taking steps to take him out of school which sounds perfect for him.
Around the house, I would lay some tarps down on the carpet so that you can at least clean up easily and protect the carpets for now. This doesn't have to be forever, just for now until you can create a routine with him. I would invest in a cmping toilet (we have a Portapotti265) and we use it with some children that are having challenges sitting on the real toilet, maybe it's a sensory issue and he doesn't like the water in the toilet or the way it flushes (when I was younger, I would be scared to fall down the toilet).
Sesondly, I would look into his diet. It sounds like he has some backed up poop issues and severe constipation, changing his diet can help with that along with giving him warm prune juice (a natural laxitive). If his poops are of epic size then it could also be painful for him to poop and this will all effect the way he wants to poop.
I would talk to your Mom about the summer and make a plan with her to do the same. Put the tarps down and explain that right now the focus is to give him as much control as possible, not to react to him when he does it and to see how she feels about the fact that he is doing this. Try and help her to feel at ease with it and to help him feel accepted and loved, even through this time.
The summer holidays will be a great time while he's out of school to create some consistency. I would really look for the signs for when he's about to go (e.g. does he go quiet? Doe he squat in the corner? Does he press on his stomach?) and then when you see the signs, explain lovingly to him that he could sit on the camping toilet and relax and poop. You can look on the internet to learn how to do an instestinal massage on his stomach which can help some children. I would also designate a plastic bowl to catch the poop to put in the toilet if it's too late. Talk to him throughout the day about how he can use the toilet and that when you poop it feels good on your body. Role model poping in the toilet for him by exclaiming when you can feel that you have a poop coming (e.g. "I feel like I need to poop, I'm going to do it in the toilet!"), then be excitied afterwards as though you have really achieved something, there are lots of taboo's around poop and he needs to know that it's a natural and normal thing that everyone does and talking to him about it, celebrating him for all his successess and giving himlots of explanatins and control will be really great!
I wish you all the best, we are all cheering you on here at The Son-Rise Program!