I feel for you OP.
What you should do, is make an appointment with your gp, you can say it is to discuss your dd. Before you go, make a list of everything you are concerned about and take it with you to give to the g.p and ask for a referral as you suspect your dd is on the Autistic Spectrum and you need support to help your dd2 cope better and also to protect your dd1. Alternatively you take the same list into school and ask the HT to arrange a referral for you.
I am a Mum to an Autistic 11yo ds and very likely on the spectrum and ADHD 9yo dd. Ds was 6 when he got his Autism diagnosis- his school did not agree that he may be autistic claiming he was doing great in school and that he had lots of friends- clearly they had never spent five minutes in his company as it only took a psychologist that long to see autism in my ds.
Girls do present differently, often they do not show signs of social issues until they are older and their friends mature whilst they don’t. But your description of your dd is very consistent with how ds presented at your dd’s Age and I do think that you would get a diagnosis for autism, based on those difficulties.
Listen to your dd, she is telling you the truth- school is incredibly overwhelming and it very likely takes everything she has not to melt down at school. Then when she gets home, a tiny trigger will result in an enormous meltdown and leave you thinking what on Earth just happened! Your dd will be struggling with sensory issues leading to sensory overload, as well as change/ transitions and lack of routine/ change of routine.
A big thing you could do to help your dd to calm down, is create a sensory space in her room- a weighted lap pad/ weighted blanket, relaxing music, sensory toys, sensory apps on the iPad. Encourage your dd to go to her room as soon as she gets in - you can give her a snack/ drink to take with her if needed. That time relaxing will help bring down the days pressures which have built up all day.
I would advise your older dd to stay out of her sisters way if she is overwhelmed (you will learn the signs such as increase in stimming and vocalising ). It is very difficult for siblings of an autistic child who is prone to meltdowns. I would try to spend one to one time with your older dd, lovebombing really helps in this situation.
I would have a look at the National Autistic Society website which is full of information :
www.autism.org.uk/about/what-is.aspx
YouTube is also a great resource.
I hope that your older dd is ok. Try not to blame yourself or your dd, she won’t have meant to hurt her sister.