Interesting development OP, things are not always as they seem! Watched some Agatha Christie on the weekend with my children and have now got my Hercule Poirot hat on!
Ranty Mum's text to Quiet Mum post panto is a massive over reaction if looked at in a vacuum.
Pre panto OP, you weren't aware of any issues with Ranty Mum and said that had she got in touch to ask about joining the after school activity there would have been no hesitation, at least from you, for her to be included.
This suggests Ranty Mum hasn't displayed any bonkers behaviour in the past as far as you're aware.
When Quiet Mum received the text post panto did she mention anything about being contacted previously by Ranty Mum?
If not, either Ranty Mum is literally deluded and has made up her previous exchanges/ relationship with Quiet Mum or Quiet Mum is not being straight with you.
If there is a back story and Ranty Mum was trying to join in the group via Quiet Mum as her supposed closest contact, then Quiet Mum should have explained the background when she received the text after the panto and not let it look like it was sent in a vacuum.
If Quiet Mum was "protecting" the group from Ranty Mum she should have explained that.
Even though Ranty Mum is not helping herself in the least and seems to have a permanent drunk texting style of communication, I'm starting to have my suspicions about Quiet Mum and whether she is quite the innocent in all this. No smoke and all that.
No matter what, sounds a bit of a mess and unfortunately you're caught in the middle OP but it's an interesting saga and got me hooked, please keep us posted on how it all turns out!