I am a counsellor and some of my clients are the parents of children in counselling. They are referred to me by their DC's counsellor. So this is my take on your situatiin OP.
I think it would be a risky intervention to bring your DS in to see your counsellor. Your counsellor has, or should have a personal investment in you and the therapeutic relationship that you have. They are there to see things from your perspective and help you.
That is what I think you and your counsellor should be working on. Ways that you can deal with your stress at this time.
Firstly, can you go for a walk and then a cuppa somewhere between leaving your dad's and getting home? You know that as soon as you get through the door your DS is going to come running to you. What you see as his demanding "something" is really his demanding you. He wants you to be his mum after your absence. Hardly an unreasonable request from a 10yo boy. How can you get back to being in the Mother headspace so you can deal with him in a loving, nurturing way between your dad's and your front door?
You say that you find your shouting at your DS cathartic. This is unhealthy for both of you and you risk your DS feeling responsible for your anger even though it has nothing to do with him. You are currently using him for release. What other ways can you release your fears, anger and frustration? Because they do need releasing. What about running, dancing, screaming in an appropriate place or sex?
Lastly, ehat would you like your sin to know about what's going on? What exactly do you want from him? This conversation needs to be had. Once you have clarity on what you want, then you can put it to him in a loving age appropriate way and really listen to what he has to say in return. This will just be the start, so try to make time to have regular one on one time with him in the weeks and months ahead when he has you to himself and you can fill him in on what's happening. I also think that you need to create circumstances that allow you to enjoy your DS.
I know that this is a very difficult time for you but your DS needs his mother and by not dealing with your own stress, you are being very unfair to him. Keep your counsellor for yourself. Focus on what you can do for you and leave your DS out of it, because it really isn't his fault or his responsibility.
I wish you all the best.
And I apologise if you think I have been harsh, I do not mean to be, but I am aware that without tone of voice it may read as such.