OP, I understand a little how you feel. My DS had something similar when he was the same age. It was an error of judgement on his part. Something he thought was funny was frightening to another child.
I was also upset that the accusation of 'bullying' was made. The two children were friends and actually the child that had been frightened wasn't the one that described it as bullying. It was the adults at the school that labelled it that.
Anyway, I had a chat with my DS, who was devastated that his friend was so frightened by something that was, in my son's eyes, just a joke. But, we talked about considering other people's feelings and what he thought he should do as a result.
What he did, was went straight into school the next morning, went to the child he had upset and apologised and said he really didn't mean to upset him and he wouldn't do it again.
And, they continued as friends, and still are to this day, several years later. My DS learnt from the incident but it blew over quickly and everyone moved on. It didn't need to be a drama. I am still friends with the parent of the other child.
But, on the day it happened, I could think of nothing else. When I got the details of the incident I honestly thought the other parent was making a massive fuss over nothing, and that her child needed to toughen up a bit. But, then, when I really thought about it, I realised that I knew nothing about the severity of the other child's fears, what impact the incident had really had, and that actually, my DS needed to take responsibility for upsetting someone else and he could learn from it.
I'm really proud of how he dealt with it, and in fact the way he dealt with it proved that he didn't deserve the 'tag' of bully, and the school agreed to remove the incident from his record.
This will blow over - an apology from your DD to her friend for the unintentional upset should be all that is needed to move on. It shouldn't need to be a drama or cause any further upset than it already has.