I've tried to back off and keep things as light as I can these last few months, but as he is so pushy I think this is just pushing DC into agreeing with whatever he wants.
What your son needs and wants is for you to step up as his parent and make the tough choices for him, rather than being put in the middle and having to watch one parent get upset whatever he does.
He has basically already told you that what he wants is for you to make the decision instead of putting him in the middle.
Backing off isn't really helping and by doing so you're denying him the chance to do anything other than what dad demands - because you're taking away any other option. It's got nothing to do with being a "pushy parent", it's about safeguarding the needs of a child with one abusive parent.
Backing off and giving dad control isn't really that helpful when dad doesn't have his best interests at heart and is just using him as a weapon. It's not that your son is choosing to follow dad's path, but that you're surrendering him to dad's control and he has lost any alternatives.
He doesn't need you to back off from parenting, he needs to feel that you have his back and will be the one to take the flak for tough decisions not him.
He also stopped supporting DC in doing tuition when at his house, and told them that they weren't smart enough to do the test.
I can't see that you've said how you responded to this. Did you correct it? Explain that dad should not have said that and it was wrong? Explain that dad was behaving badly because he wanted to upset mum, and you're sorry that dad hurt him instead? Remind him it's not his fault this is happening? Do anything to rebuild his confidence? Tell him that you believe in him?
Or did you brush it away / minimise / make excuses to try and avoid 'conflict'?
Bigger picture aside, I'd be concerned that giving in to cancelling tuition when he's with you, giving in to not sitting the exam, backing off and letting dad drive what happens - you're basically communicating that you agree with dad that he's too stupid.
"We're still going to do the practise and sit the test, because I want you to have lots of options open to you. Dad was wrong to call you stupid, because it was a hurtful thing to say and it's not true. When you're with me we will do practise together and when you're with dad you might do different things. That's ok."
Or whatever, but that's the kind of messaging he needs to receive from you. Along with praise and encouragement.
Big picture, I agree with pp that he should sit the exam. By not doing so it closes down his options and hands his abusive dad additional control/power.
He probably won't thank you at the time, no, but you have to be clear and open with him. Explain that you are making this decision because he is smart and capable and you want him to have as many options as possible. Explain that you will tell dad you have made the decision and you will deal with it. He is not in trouble. Be consistent.
If he is distressed about tuition you work on restoring the self-esteem his dad has trashed.
For him to feel able to make his own choices even if they are different to what his dad wants, he needs to feel safe, secure and protected. He may also need more support to see where he has choices and to have a framework for decision-making based on what he wants rather than what he thinks dad wants - that's the consequence of being controlled through coercion and he will struggle to communicate his views in a court process without those skills.
Give him lots of opportunities at home with you to make choices about small things that have nothing to do with dad. Ask him what he liked or didn't like about things that are happening at home. Ask what he might like done differently.
He might not know how to answer those questions at first but keep giving him those ideas to mull over. By asking them in situations where he has choices it will help him to learn to recognise the link between the situation and the idea of choices.
But all about small trivial safe things. It's a way to help teach him to recognise what he likes/dislikes and what he wants/doesnt want, as well as to be more able to recognise situations where he has choices he could communicate. As opposed to now where nothing feels like a choice, it's about doing what dad likes/avoiding what dad dislikes, and therefore he struggles to recognise when he should have choices or what to do.
Slowly but surely you get to teach him skills (and confidence) to overcome and break free of dad's coercive control without the conflict. That will set him up to one day feel able to say "no, I don't want to see dad today because he treats me badly" - rather than feeling he has no choice but to see him. It will also make any court processes where he is asked for his views more manageable for him.
Learning how to understand when you have choices and how to make them is a really tough thing to learn after being subjected to coercive control, especially for a child. But you have the power to do something about that.