Personally I just have huge sympathy for him because I have a young adult with ASD myself. He's lashed out by repeating a horrible word that everyone at his school uses because he's been horribly mocked - from an autistic perspective why is it ok for everyone else to use that word but not him? The girls ablism is at least as bad as his misogyny - and she started it. You say he said it to his friends, did they tell her? If so he's already feeling attacked by his friends, not to mention by the girl, also by school because he hasn't been able to defend himself - this is a kid who needs someone on his side, and that person needs to be you.
IMO opinion writing a letter is an absolutely terrible idea, that is just asking for him to be further humiliated. Do you think this nasty girl who took the piss out of him being autistic is going to appreciate the letter? Do you think she's going to suddenly develop empathy and start being nice because he's apologised. No of course not, she's probably a nasty kid who will laugh at the letter, show it around to all her friends and use it as more ammunition to humiliate him. I cannot imagine any worse idea that this for your poor kid.
Cancelling his party next week is the next worst thing you could do. This is a kid that probably struggles terribly socially, has just been mocked for his disability and you want to cancel his party? That is madness. Talk to him gently about using this word, be clear it's not ok and that some silly girl is not worth him getting into trouble for. Be clear about him not using it again and how he can handle it differently if something like this happens again. He needs support now, I expect he was terribly upset by what happened and I personally don't blame him one bit for lashing out in return.
He has ASD, he can't put himself in someone else's shoes and his emotional maturity is years below his age. You need to take a different tack with this kid, understand where he is coming from. IME it is not unusual at all for a kid with ASD to think that because someone else says or does something that that makes it ok. With the racist jokes he may have heard someone tell them and other people laughed - and that's what he learnt, if I say this people will laugh and think I'm funny and like me. He might not even understand the joke (I don't know) but people think it's funny so he's saying it to make people laugh and he can't see beyond that.
Please don't feel ashamed of him, just spend a lot of time reading and researching autistic behaviour and why he might behave the way he behaves. He needs you really badly OP because secondary school is probably a really, really tough place for him. Concentrate on other ways for him to handle things which will help him in the future rather than focusing on punishing him which will just reduce his (probably already rock bottom) self esteem. Role playing examples and going over and over and over it may be really helpful to him.