I understand why you want support, but when you blend families and your new partner has children, you do knowingly enter a situation where you are aware they have DC and your partner will have to put them first. It’s a hard situation for you if you are struggling with mental health problems, but he is also between a rock and a hard place. In his situation would you want to destabilise your relationship with your DC when things have been tricky? You aren’t just asking him to compromise- you are asking his children to. Maybe he worries it’s one day, as a one off now, but then this sets a precedent or you will want to keep that change if it helps you.
As it happens, I don’t think this situation is sustainable and I’m not sure how it can be resolved. I think your DH needs to explore some family therapy with his children first, then potentially involving you.
My other concern here, OP, is that all the “blame” is being placed on your step-children. I have no doubt their behaviour was an issue, but I also believe that will have come from somewhere- and I am certain that “somewhere” is at least in part unhappiness and feelings of insecurity. You appear to almost resent your step-children for being the problem and making life harder for you- but, honestly, this sort of behaviour does not come from absolutely nowhere and without an underlying reason. It just doesn’t. All children can be naughty, argumentative, will try and wind up or annoy siblings/step-siblings or play parents off against each other- but for it to have got to the point where they can’t be in your house is a huge leap on from that. I think you and you DH need to reflect on what else might have been going on for his children. Are you sure you/your children were not contributing to the issue? It is rare one side is always in the “wrong”/misbehaving and the other always behaves perfectly when it comes to conflict.
I suppose that I don’t believe things were “perfect” in the household before some photos of their parents were shown, given that they have no memory of their parents together. if all was well, some photos and memories would not have caused this. You might have felt they were perfect- and from your perspective, they might well have been- but perhaps your step-children weren’t quite as happy as you thought? Children can be quite good at masking unhappiness and playing along, especially if they feel it will gain parental approval and that is something they crave, or feel uncertain of. Sometimes, things change as they get older as they feel more able to act out/voice issues, or work out alternative ways to get their needs met. They may have been jealous of their father spending more time with his step-children than them, for instance. It might not have been that, just an example.
I think you are at risk of demonising young children as the problem. I suspect they aren’t actually the problem, they are merely acting in response to the actual issue. I also doubt that you and your children In any case, creating a situation where his contact with them is lessened or disrupted is highly unlikely to help matters.
Hopefully, with some professional help your DH and his DC can work through it, before involving you and working towards a better relationship with you too. I hope with professional help your mental health will also improve.