Oh Dung, that's a crock of shit ! Good to get it off your chest, and it makes perfect sense.
So glad you were honest with your dc about their dad's behaviour. It validates their feelings, shows them they don't have to think it's ok if their mum, an adult, also doesn't think it's ok.
Can I suggest you take the initiative and email him first.
You've marked in your diary that he'll be taking dd to her club that day in November.
His request to switching dates for your birthdays. No. And no explanation.
Tell him dd doesn't want to go to his Tuesday. No explanation.
Advise him that next time he goes to the cm she will be expecting him to pay her, as that's what you've told the cm.
I don't know the details of your situation so some or all of the above suggestions may not apply. But to an outsider that's how I'd see going about arranging things, but not engaging. No digging, no point scoring, just facts.
Don't email just now. Calm down and do something nice if you can. After you send the email don't engage with any reply, whether it's a call/text/email. Ignore, ignore.
And remember. Your dc are where they really want to be - with you.