I empathise with this a lot. I'm sorry you're having to deal with it. My DP has some deep rooted and complex mental health issues and it's thrown up some challenging behavior at times.
My chat to him over the years has always been that mental health issues aren't his fault but they are his responsibility. I've had to set clear boundaries about what I consider to be acceptable behavior and what isn't and you can do the same. It's not unsupportive at all. If anything, it's the opposite because you're making sure that your relationship can survive the situation and that he's not ultimately going to be left in a worse state on his own. You obviously have a lot of love and empathy for him so I doubt you'll be setting any boundaries that won't be understanding of his limitations right now.
He might need help to get underneath his emotional state but he also needs to do some work on his behaviours and habits too. Some mental health supports are really good at helping you unlock all the reasons why your head's a mess but don't necessarily leave you with the tools to make change. Some go the other way. Whatever form of help your DH is getting, he needs to be an active participant which means more than just going to a session and talking to someone. He also needs to apply what he learns and make a commitment to recognise the wider impact of his behavior on your family. That might take a bit more time when he can't see the wood for the trees yet but you can hold him accountable while still being supportive if you approach it right.
I've had to spend time with my DP helping him identify his triggers and reactions because I've learned to spot them better than he has. I usually try to keep the conversation really gentle and understanding along the lines of "I'm noticing this... Have you noticed it? What's happening there?". I don't expect him to never slip up or do it again but I do expect him to engage with me.
Sometimes he's not as receptive and tries to excuse the behavior with the underlying reasons. I always take the view that the reasons don't change the impact. The hurt caused is the same. The reasons are what makes me forgive and move past it but he needs to understand that his behavior had an impact.
And sometimes I just lose the rag and tell him to fuck off. No one's perfect and you don't have to be either.
I hope things get better. He might be on a long journey with this or it might be something he can get over quickly (hope so) but, either way, you are absolutely allowed to have boundaries. He has to play his part in his own recovery and, being in a relationship and a family, he needs to accept that he's partially accountable to you guys for the sake of everyone's wellbeing.