Your second post shows more unusual behaviour. I would lead with this kind of thing. It is less likely to be easily passed off by a busy gp as you looking for an excuse for poor parenting.
Try to spot triggers for the other less acceptable behaviours. Loud noises, bright lights, certain sounds, certain clothes or activities.
Then mention the triggers as causing the behaviours rather than focussing on the behaviour itself. This will also help you to manage it, as you can divert away BEFORE a meltdown, and warn teachers to keep her away from those situations, and teach her to recognise her reactions.
I've taught my 4 yr old to politely say "I'd like to play alone please" instead of screaming "go away, don't sit near me" when she needs space. She's very proud of herself when she says this, as she finds it very hard to talk to other children. Better with adults tho.
Do lots of research on subtle symptoms in girls, style of play, ask teachers if she does eye contact ( I thought my dd was fine cos of her excellent eye contact, turns out it was only with me...)
Is there anything repetitive (speech or activities) , restricted, sensory.
Eg, same routes/ routines every day, food refusal (mixed food, or specific textures) needs labels cut out.
A year on, I still look back at my dd of two years ago and think it's so obvious in hindsight, but at the time, she just seemed quirky.
I brought a 6 page typed letter with me to the paed, prepared it over the two months I waited for the appointment. It was grouped into social, language, communication, physical, and about 7 other sections. I just kept a note open on my ipad and added to it whenever I thought of something that seemed unusual, no matter how small. When I read it back, it was blindingly obvious.
I suspect my overly obsessive preparation was seen as another tick box
Genetic history of asd? Tick.