Please try to be gentle with me here because I may have fucked up but I haven't intended harm. And apologies that this is long.
I'm trying to be as objective as possible here but also give relevant context.
I have an old friend (20+ years). She is a very dear friend who I've been through a lot with and she has been very supportive and will go the extra mile but is prone to being a bit of a diva and has form for big arguments with people who she perceives are not treating her well -- she's fallen out with several people in similar circumstances.
I work full time in a v stressful job and have a DC and almost no childcare. DD's dad, who I'm divorcing, is barely in the picture, doesn't support me in any way (financial or practical). It's extremely difficult for me to get out of the house - I basically always have to use paid babysitters.
Friend has no DC and doesn't and never has had a full time job. She does work but in short, intense bursts with long periods without work, and she doesn't really get the stress levels inherent with office jobs and the commute that goes with them.
Friend and I see each other as much as possible -- we last saw each other about six weeks ago which admittedly is a fairly long time for us.
I cancelled on her two weeks ago due to illness. She subsequently cancelled on me the following week, also due to illness. Roll on last Saturday: I had been out the previous Friday (long planned). When the last of the cancelled meetings got cancelled (by her), we arranged for her to come to mine on the Saturday -- I knew I wouldn't be able to get out on that evening and offered to have a catch-up with her. My DC had stayed over on the Friday with good friends and was tired and specifically asked if we could have the Saturday together, just the two of us.
I texted friend on the lunchtime of that day, very apologetic, and said would she mind if we moved it because my DC had asked me to and I felt I had to respect this. She could have come after bedtime but in reality it would have meant not until post 9pm.
We then had a drink in the diary this week with another, mutual friend. I messaged both of them on Tuesday to check it was on. My friend replied (to both of us) that she had other plans, which we were a bit 
about but was left. Me and mutual friend agreed to reschedule.
My friend later texted just me to say she'd deliberately pulled out of the drink this week because she was so furious with me for cancelling on her. I followed up with an apology which was a bit half-hearted, making the point that I don't like cancelling on people but due to my circumstances this is sometimes unavoidable.
Friend has now gone absolutely nuclear and sent me a text yesterday saying she is at the end of her rope with me etc.
On the one hand, I really regret that she feels like this. I understand that its shitty to be cancelled on twice, and I feel I owe her an apology. On the other, I feel that this is an over-reaction on a fairly massive scale which is childish and a bit controlling. She cancelled me once so is at least one third responsible for the fact we haven't met up, and then threw a tantrum affecting an innocent third party in order to make a point. While it is incumbent on me to do my best not to let her down, I think she needs to understand that due to my situation these things will happen from time to time and that I shouldn't be made to wear a hair shirt when it does happen.
Every day is so stressful for me that I just feel I don't have emotional space for this kind of friendship any more. We're in our 40s and while I really value old girlfriends and move heaven and earth to keep up with them, I don't want to feel that things like this which happen to people all the time are going to lead to a huge emotional meltdown, a bollocking and a gigantic reckoning every time they occur. It feels unrealistic, childish and a touch controlling and I can't be bothered with it.
I know I won't be able to move past this with her without a row, a long, tear-filled discussion and many apologies on my part (which I don't really feel like making). For now, I'm just leaving it. But if it comes down to rescuing this friendship on her terms or refusing to be dragged into the drama, what should I do? Has it run its course? And is this why I'm not feeling hugely apologetic? Or am I being a selfish cow?