This is very upsetting and frustrating for him, I can see he gets visibly upset and angry and this often results in an argument between him and her (which undoubtedly is what she wants to achieve!).
A high conflict ex is never helpful to a new blended family. Your partner needs to not argue with her (because they are both being high conflict then). If she’s being difficult about contact, he needs to take her to court so she can’t. It shouldn’t be disrupting your life.
I am in that parental role when we have the children to stay with us and I try my best to make their time enjoyable with activities they like and make an effort to get to know them. I'm mindful of the short time that my partner does have with them, so I do what I can to take on the boring day-to-day stuff like cooking and cleaning/tidying up so he can spend quality time with them. This often means that I feel those weekends for me are just house work!
I know that you are trying to do your best here, but this is going to lead to resentment. The children are there every second weekend (which is what he’s agreed with his ex), but life still goes on when they’re there. They should see their father doing housework and doing to boring bits of parenting. Don’t become his skivvy so that he can spend his contact time with his children because you will become resentful that half of all your weekends (so the time you aren’t working) are spent as a domestic drudge.
He should be contributing equally to the housework all the time, not just when his children aren’t there.
Even more so if he then isn’t making the most of his time with them. If he’s actually getting leisure time and not doing things with him. You’ll find yourself resentful about this. No parent is ever going to be full on attentive the entire time - but, even knowing this, you will resent him pissing about on his phone rather than playing with them if you’ve taken on all the crap to give him the time with them.
It sounds like you’ve taken on loads of the mental load here - you are planning activities and such like. Don’t. He needs to be doing it. For them and him as much as you.
You are going to burn out and he’s going to think he’s entitled to all your labour if you aren’t careful. That’s not a good dynamic to cultivate.
This might sound odd, but what’s driving you towards that dynamic will in part be the ‘his previous, limited time with his children’ narrative. You need to challenge this. He was involved in the process of negotiating a contact arrangement for his children. And he agreed to this. So he needs to accept it and not frame the whole thing in a poor him narrative.
I try my best to support him with the fallout from the awful dynamic between him and his ex, but the negativity of it is affecting my own wellbeing and mental wellness. The level of contact from her seems relentless and far too frequent given that he only has them for four nights each month really. It's like she's a constant background feature in our daily lives and that makes me feel uneasy and sometimes actually anxious.
I know this feels like an ex problem. But it’s actually a partner problem. He is perpetuating a high conflict dynamic with his ex and allowing this to take over your life. He doesn’t have to. He needs to get his house in order and stop making you miserable with it.
I tried to speak to him about this but it's resulted in ugly shouting matches
This a bad sign. He shouts at his ex. He shouts at you if you try to raise the ways this is affecting you.
You are feeling anxious, upset, overwhelmed and resentful because, frankly, he’s giving you the shitty end of the stick. It’s not ok.
You haven’t been together that long. Honestly, I’d suggest that you move out if possible and try to reset the relationship.
Step back and leave him to parent - and do housework - during contact. Draw clear boundaries: His conflict with his ex is not your concern at all.
If you do choose to go to counselling, make sure you choose a counsellor with considerable experience of stepfamily issues. Because most of them do not understand the dynamics at all, and they can often make things much worse.
Maybe have a read of Stepmonster. It’s enlightening.