Wow. This isn't about the safety of a horse riding party at all. It's about this other mom. It seems her concern and desire to have grandma attend is the last straw for you in a very long history of just not liking this woman and, by the way, not "approving" of her or her family or her choices, as in:
You make sure to mention that you have a "very stable home" (you somehow feel she doesn't, or she has criticized your home indirectly as being somehow unstable?). You throw a nice party "so it's not like we're the dregs of the area" (did she say you were? Why is this even mentioned here?) "The aftercare they use is pretty lousy, every (one?) else thinks" and "she's fine with this lousy aftercare program." (Who is "everyone else" and why do their opinions matter here? Maybe her family can't afford or get a slot in the more desirable aftercare of which "everyone else" and you would approve? Why is this family's choice of aftercare even a subject for discussion among "everyone else"?)
I do think she was over the top raging at you about the pickup she didn't expect. But did you take a second to think about how she might have been scared to death when she realized that her child wasn't where she thought (even mistakenly) that her child was supposed to be? Wouldn't you feel the same for even a moment in her shoes? You were acting based on an honest mistake or miscommunication and she should not have reamed you, I agree.
But even if she's a jerk, why are you so emotionally invested in her and her choices? She isn't asking to be your best pal, she's interacting with you related to your children. So she has an issue with the party. She is free not to send her child. Tell her the daughter can come but you've reassessed seating in the car, have to bring other kids, and there is no room for you to drive grandma as well. She likely will pull her daughter from this party, and will be done with you as you clearly wish to be done with her. The daughter will be the one to suffer as she will be hurt and disappointed. But you'll be rid of this person you clearly don't approve of or like.
You seem to feel she doesn't trust you and that angers you, but why do you care so much? If you cared because you were close friends with her and she still lacked trust, that would be different. But it's clear you aren't friends and don't like her or her choices, so why does this get under your skin so badly?
Your SWH claims "when did I say I can't stand the family?" You did not , in so many words, but there are plenty of phrases that show you don't approve of their choices or feel somehow slighted by them. So let them go.