I am a parent to two children with ASD, both of whom have suffered horribly at the hands of bullies.
I am NOT part of the hang 'em and flog 'em brigade.
Permanent exclusion from school is a very serious last resort and it doesn't actually help, it just moves the problem elsewhere. It can ruin the education of students who have just gone astray or made a poor judgement. There will be times where it is necessary, but in all but the most serious of assaults, it should not be considered as the first step. And victim's parents don't get to make that demand. (well they can make it, but the school will ignore it)
Punishment is important. Fixed term 1-2 days exclusion is appropriate. But what happens after that is far more important. They need to be taught how to not bully and how to not stand by when someone is being bullied.
People bully because they want to belong and be accepted. They will do anything they can, to separate themselves in their heads, from their victims because the thought of being in that position is scary, and it's always better when it is someone else. It's just like on this thread where some people have jumped on (and bullied) the OP and blamed her for her DD's behaviour. They know that their DC would never do something so awful, because they are better parents. If they can blame the OP, it reassures them that the same thing could never happen to them, and if it did, they would come down on it so hard, it would never happen again. Oh so simple and oh so deluded.
You can't make someone be empathic, they need to learn proper understanding. They need to understand what it is like to be isolated and treated differently. School can use Drama or PSHE to teach that. They need to teach kids how to stand up and say stop, when they see a child being bullied, even if they are saying it to their friends. It's hard, because most people don't.
I would spend some time with your DD, talking to her and asking her, what she thinks it must be like for this other girl. Ask her how she might feel in that position (A proper calm discussion, not a shouted 'how do you think she feels!'). Ask her what challenges this girl must have, and what she might need from people around her. Ask her what she might be able to do if she found herself in a similar position again.
I hope you make some progress with her.