Dear M.:
Been there, done that.
I would stay out of it. It sounds like the divorce was not too long ago, because such behavior should cease within a year or two at the latest. Your boyfriend is acting very appropriately. He remains courteous and lets her know that such behavior is not acceptable.
It would help to know their decreed roles, like who is the primary custodian. You COULD turn down her request, but I would always try to accommodate the child. In four years since my divorce I turned down only two days where I had plans I was not willing to change.
When I was in the phone/yelling situation, it told my ex-wife calmly and only once that I do not tolerate cuss words and yelling, and that I would hang up immediately. You cannot reason with infuriated people anyway. Since she only called when she wanted something, my hanging-up policy worked within days, and although I do not like her at all and keep the conversations brief and to the point, we can talk calmly now. Not yelling back reduces the amount of anger and ammunition. I will gladly be a boring, monotonous, cold person to talk to, if it buys me the shortest necessary exposure to my ex-wife.
I would not even question the reasons why she reschedules, just say yes or no. However, with a few exceptions I also stick to a 24h rule, meaning that I want the notice early enough to change plans without standing up someone else.
On the bright side, things get better in time. However, your partner's 'ex' (or 'current' from the other's perspective) is not one you want to get close to, if bad vibes are involved on any level. And even if not, you may be better off staying away - other than hand-offs of the children.
That's my 2 cents,
W.