OP - my heart goes out to you as I've been in a similar situation (not as a single parent though so you've got so much going on)
The school and local authority have a duty of care to your daughter and to provide her with an education. I'd suggest trying one last time to get the SENCO & Headteacher to support you, then explain that you'll be forced to make a complaint to the Governors about this.
It's all very well the school saying she needs a specialist setting, but in order to get into one you do need a body of evidence and the school will know that.
In the meantime, would going into school a little bit later help your daughter? I did this with autistic child and it really helped just being a bit quieter, with less pressure in the morning. I know this might be tricky around work though.
I'd also suggest getting in touch with the Autistic Society for advice about transitions and ways to help her get into school positively. They run an "Early Bird" Parenting course that has helped me and my family so much in terms of understanding our child.
I'd also recommend Autistic Not Weird on Facebook & Autism Happy Place on Instagram (both run by autistic adults, who give great insights into how your child sees things)
Another thing we do is to make a visual list in the mornings (get dressed, eat breakfast etc etc) We tick it off and it helps them to know what needs to be done, and in a calm way.
We actually have one programme in the mornings (Hey Duggee or Bluey, my child's choice) and it's really helped to calm them. I think screen time usage looks different when you have a SEN child - not saying let her what as much as she wants, but a little bit here and there is good to help them regulate.
I limit questions and chatter in the mornings, even sometimes just leaving them alone for 10 minutes in quiet really helps.
Lastly....just sending you a huge hug and some solidarity. You're doing brilliantly, especially with such an unsupportive school.