He says he feels differently to you about this and he doesn't want you to be involved, and I think you have to listen to this. If your friend speaks to her son about it, it could make it feel worse from your DS perspective. I think you were absolutely right to step in about the bullying, but this is different.
Your friend's son has accepted invitations from the bully and made new friends. That isn't loyal, but I don't think teens always are and nothing his mum says will make your DS seem like a more appealing friend: possibly the opposite. I think you have to trust him to navigate this. Be his support, listen to him but keep his confidence. I have fallen foul of this myself. What your teen tells you is not what they want the world to know about them. He might be handling this with a bit of nonchalance at school. You don't want to blow that for him.
Bullying is horrible, wrong and action must be taken. Not inviting someone to hang out, no matter how pointed, is not something a mother can fix. Trust him. Of course he doesn't want to change school. He probably does have other people he gets on with at school and other friendships will grow.
I have been on both sides of this:my DC was left out of a group which was horrid but it resolved and most of those involved are now her school friend group, and a friend of mine think my daughter, plus this friend group, are horrible to her daughter. The side of the story I get is that this girl hates the rest of my DC's friend group so they don't hang out at school. Things my friend has said corroborate this. Her child barely speaks to my DC some days, despite my DC being polite.
I cannot have a conversation about this with my friend. She was very upset and I think she now dislikes my DC, who actually hasn't done anything wrong. They were never close friends at school. Now my friend's DC has a gang of mates, things are better but there are still no go topics of conversation, which is horrible. Her DC has only given her perspective and my DC has given hers. The truth is probably somewhere between but I can't command my DC to leave her friend group to befriend this other girl. You can't choose their friends for them. If only you could!
This is the hardest element of parenting for me so far, but my friend and I are much more upset about it than the girls ever were. I think that's key. We put an adult spin on it. Teens are awful to each other one moment and reconciled the next. Ask your DS for positive school stories. I bet he's got some and you might feel reassured that he's actually dealt with this quite well himself.