'I can't leave him for reasons I don't want to go into.'
I'm sure you can. If you share the reasons with someone- such as here, someone else you know, or with a counsellor, they could help you find a way.
'But what can I do to support my children and to continue to strengthen their mental health.'
Leave him.
'he's at home now a bit more than usual too, which the kids and myself hate.
I've spent years upon years walking on eggshells with him, its just horrible that he has more time at home now.'
This is profoundly damaging to your children's mental health. They're saying/showing they hate him being around because of how he makes them feel. My sister and I grew up with a dad like this- walking on eggshells and living with a volcano. We've both with left with anxiety issues, because we spent 16+18 years respectively with nowhere where we could relax. Then our mum left him, and we saw her for the person she was. She's a very kind and helpfuul person, but when she was with him it didn't show at home, because she let him do all the name calling etc or whatever, and didn't stand up for us.
You know what you need to do for the well being of yourself and your children. His behaviour could effect their health for life, as well as their ability to work, and to have healthy relationships (it might even have all these effects on your own life, too.)