Op, this is a very tricky situation. I agree with other poster that this could possibly be an age issue and boundaries being pushed.
However, yr comment "Dp has been quick to lose temper and shout recently. Saying he can't go on anymore and he's going to move". This is not proportionate and if he did leave, he would also leave his DD so i think there are other issues here at play. How old is yr DD?
Also, a few of yr comments such as: " I felt very grateful for Dp in our lives" and " "He's always provided for DS" are a bit worrying. You should not ever be " grateful" for anyone. Also, provision for yr child was never yr DP's responsibility although, i appreciate he has done that.
I suspect yr DP enjoyed family life while it was good and easy but not so much now , challenges have arisen and reality beckons.
"Saying he can't go on anymore and he's going to move."
In this case he is either:
Genuinely unhappy- let him move.
Or
He is struggling with the reality of parenting 2 children and threatening to leave - let him move.
Either way he is not happy, he is shouting and, your DS and DD both deserve a peaceful home in which to grow up in and be themselves. I assume you face the same challenges daily but don't threaten to leave! He clearly feels he can up sticks when it suits him and the going gets tough, so let him go.
He is somehow slyly suggesting that you reign yr son in, so that he ( DP) has a stress free life or he will leave. You cannot do that and your son deserves to be himself as he is growing,understanding and challenging this world. Your son is becoming a young man and finding his feet in this world. He needs parenting, not shouted out and " Dad" may leave!
Let him leave and go live his best life. He is really no Dad or Dad sub. It takes more to raise a child than a few quid each payday and collecting from sch. You are mum, you are your children's protector and defender. You don't get to leave.
Help him with his bags and make sure he doesn't trip on the door step on the way out. He does not get to blackmail you or your children. Take that power away from him, let him go and allow yr children a shouty free , loving, all encompassing home with you.