Sorry in advance for the long response. I just really feel for your situation.
I can relate, big time. My last long-term partner had (undiagnosed) MH problems and basically behaved the exact same way your partner's ex is behaving. My ex had a daughter (my then-step-daughter), and I felt like the only adult/parent in the situation. It was maddening and extremely sad. But, like you, I had no legal or actual control over important things. I was basically swimming against the tide. It ultimately came down to me drowning in it, or choosing to get out of the water.
You're in a messy, dysfunctional, seriously troubling situation - and in all honesty, I don't know how much you will be able to do to change it on your own.
I can massively empathise with BOTH you and your partner.
I've been where you are - unable to have any actual say in things, yet also being the only one actually trying to parent the child in a healthy way.
But I've also been where your partner is - dealing with someone who isn't capable of looking after themselves, let alone a child; and dealing with the exact same things - such as being blocked, being threatened, prevented access, etc. It's one of the most painful experiences a parent can go through. It's a primal thing. When access to your child is removed on a whim, you begin to feel absolutely desperate and you start placating out of pure fear. That's what your partner is doing. He sounds terrified, honestly. He'll be going through his own issues dealing with his dynamic with her, also. He'll be triggered in ways that you can't see on the surface, but that are evident in his avoidant/passive behaviour.
Ultimately, you're NOT wrong to be reconsidering your relationship, nor to communicate that. It's completely understandable. Something needs to change.
If this was a one-off argument, then the two of you need to sit down - when you feel better - and really talk this through. You need to listen to him and hear him properly. You need to ask what he actually needs from you, and what he actually feels. It's possible that everything has just been unspoken and simmering away, and now it's come out in an unintentionally nasty way. Maybe he feels that your frustration with his ex's terrible parenting comes across as frustration with your step-daughter (and her behaviour). Just listen to him. I'm not saying he's right. Just to hear him. It's also possible he's equally as disordered/problematic. That's for you to determine, really.
For you to stay, he also needs to listen to you and take you seriously, because he has to understand that he's not the only one affected by this. Perhaps hearing how you feel can be a catalyst for him to really step up and get this sorted. If not, I don't see how you can continue.
The final thing I would say is that I think in either case, it would help you to step back from the situation for a moment and realise one very important thing:
You will NOT get any logic or sense out of your step-daughter's mom. You absolutely 100% cannot expect her behaviour to be reasonable. It won't be. So if you are going to stay in your relationship and continue dealing with her, you will have to alter your perspective. Any time she flies off the handle, says something ridiculous, fails as a parent, threatens, withholds access, blames you, behaves in a dramatic way, etc. - you can't get sucked into the details of it. Just know that she's going to be irrational, because of what she's struggling with, herself. Don't try to reason with her, because it won't work. You'll only drive yourself insane.
There are only two healthy outcomes for you. One is that you leave, and report your step-daughter's mother on your way out. The other is that you stay, but something seriously changes. Your partner needs to have a way of legally having access to his child - as in, a court order or some such that doesn't allow his ex to prevent contact whenever she feels like it. That's for a start. In all honesty, I don't think your partner's ex should have custody at all.
Get better soon OP. Good luck with this.