Forgetting to make an effort for your birthday (and have you pay for your own birthday dinner) was callous, and he deserved being called out. Then over-dramatising the situation and himself getting (known to you) police involved for you having a go and screeching that he was selfish - what an utter over-reaction! There was no physical violence, no name-calling, and you guys rarely fight, so no prehistory of things potentially turning violent. The only reason why he did it was to shut you up, to embarrass you publicly (and punish you for calling him out). You know what could have stopped the argument in its tracks? Him admitting and apologising that he had deprioritised you and should have done better. That would have stopped it. But he wasn't contrite, he just wanted you to publicly suffer and be embarrassed for calling him out (and waste police resources).
Then the on-off drawing you in and pushing you away with the communication since he left. 'Don't talk to me', but 'this is a picture of us having fun' and 'join us at karaoke', etc. This is specifically done to make you question yourself, and to keep you mentally occupied. Again, it is manipulative and punitive, and will make you question your sanity. Moreover, with you engaging with him (based on his few positive interactions with you), he can call you out for not respecting his 'boundaries', again allowing him to paint you as abusive. Only communicate in writing from now on in one medium, so you cover yourself. Keep it short, factual, and only child/household admin urgency based.
Then the 'you must be drunk' in a genuine accident in the pub, while he knows you aren't (and apparently considered you still capable enough to supervise his daughter).
From where I am sitting this does not sound good. This man is not acting like your best friend and love of your life. He has been escalating this situation at every opportunity, where de-escalation was possible, and has consistently attributed the blame publicly to you. And has successfully convinced you, that you are at fault, because you got emotional. You losing your composure twice is just shrapnel in the war of attrition he is waging here. And you are here, being all regretful and sad, just hoping to turn things around, ignoring all these alarm bells.
This man is 100% manipulating you, whatever his agenda is. You cannot trust him and should second-guess his actions and words. He is painting you poorly in both your communities (your circle of friends will undoubtedly soon be informed too of your 'transgressions'), and in front of your kids, and the question is for what purpose. Please lawyer up, and start seeing the bigger picture here.
Your relationship is broken, but not for the reason you think it is.