I've just read through all of your responses OP.
There is no surprise that F1 wanted to handle this without the involvement of the other mum if at all possible.
I have been F1 and dealt with an F2. I did the opposite to F1 and asked to sit and talk about with when it first started. My F2's response was that her child couldn't possibly be a bully and reeled off lots of instances where the child had been praised for being a kind person. F2's child is the type that sucks up to all the adults around her and is slyly horrible to the other children.. So, when this was starting and I wasn't aware, my daughter had a sleepover with this child. I was out for the evening, and my daughter didn't have a phone but did have her ipad and messaged me that she wanted to come home because this kid was being vile. I didn't see it until the morning and rushed up. My F2 was incredulous and said they'd had a great time and actually said that my child was manipulating me because she didn't want me to go out.
When I got my daughter on my own I asked what had happened, and she said it was like a switch. When her mum was there she was all sweetness and light and solicitous, and as soon as the adults left, she was bossy and controlling and telling my daughter she had to do what she said, as it was her house. Made her do lots of things she was uncomfortable with.
When it all blew up, F2 spent hours talking to our mutual friends about how upset she was. About how her child had always been so kind to mine, and how much she had done for me too. That she couldn't believe we would make up these stories and how awful we were. Does this sound familiar?
Anyway, F2's daughter has since fallen out with multiple other kids in secondary and really struggles with friendships. She's bossy and controlling and other kids don't like that. F2 has also fallen out with other mums.
No-one likes to hear that their child is a bully or being mean. But the difference is in the parents reaction. If something told me something like that about my child (and they have) my response is "I'm so sorry, I'll talk to them and find out what happened." I do not believe that my children are angels who can do no wrong. All of us can and have behaved badly at different times. It doesn't necessarily make us bad people, what affects that is the reaction around you. If you are constantly telling your child they are magnificent and the best person in the world and it's other people in the wrong then it is doing them no favours. I suspect this is the case with your F2.
It's interesting that your F2 is also very involved with the school, my F2 was the head of the PFA. It's definitely a certain type of parent! (I say that, having also been involved with the PFA).
I understand you wanting to stay friends with both of them. However, do ask yourself, how would you feel if it happened to your child. How would you want mutual friends to react.
We all think that the bonds made at the school gate are strong ones, and we see facebook posts of people going off on holiday together long after their kids have grown up. But that's the select few. The reality is that they are often fragile and ephemeral friendships, that don't stand the test of challenges like this.