Oh God. You sound exactly like me a few years ago. I tried everything to appeal to his sense of responsibility as a parent, tried to make it easier for him, ended up doing loads of stuff because I couldn't bear to see DS suffer XP's horrendous snappy moods. No child deserves that.
I'm going to be blunt; you may not be ready to hear this yet but hopefully at some point you will. He's never going to change. In fact, he will probably get worse. This type of behaviour springs from a deep-rooted and unconscious attitude of entitlement that will never go away. I know you really want to make this work but you are never going to be able to, not with this man in the mix. It just gets worse and worse and you can't MAKE a person change.
The only, only way they will ever improve is if, independently of anything you do or say, THEY recognize they have a problem and seek help themselves. Such men have to be really motivated to change and even then the habits of a lifetime are hellish hard to break. The percentage of such men who actively seek treatment because they realize there is something wrong with the way they are behaving is tiny. Of those, not all succeed in changing. From what you have said (1 'token' visit to the counsellor) your DP is not one of this tiny percentage.
I really wish I'd got out when my DS was small before DP could fuck him up too much; it DOES affect them even at a young age and you can't protect your son from his horrible moods if he is living in the same house as your DP. You just can't.
I left my XP in the end; it took me three goes, once when I was pregnant when we stayed living under the same roof, once when DS was 7months when I moved out for a month or two and then went back, defeated by PND, and last and third time was the charm, when DS was 20 months. I wish I'd left him sooner. Now me and DS live in our lovely little rented cottage, I work from home and look after him, have just finished my OU degree and am very happy. Right now DS is making mess with glitter and glue in the kitchen with his granny and there is no-one scowling, glowering, huffing or slamming out in protest at the disruption to his tightly ordered world. DS sees his Dad 3 times a week in the day and comes home tired and subdued, but he is so much better off than if he got to see me constantly shamed and talked down to by his Dad, day in, day out. He has a happy mum and his Dad and I are civil to each other. I have a lovely new DP who makes me so happy and treats me really well, does all the right things. The contrast is startling. The thing is, I received better because I DEMANDED better, and I could only do that when I realized I deserved better. That took me a long time. If you saw a counsellor on your own it might fast-track your self esteem a bit so you could learn that the only acceptable way of dealing with blokes like this is to hightail it out of there, because they will never never change but you can change your life and be happy, and you should :)