The people who question the reason to go for an assessment because a diagnosis won't change anything are really misunderstanding themselves. If a diagnosis wouldn't make the slightest bit of difference to someone's life it's highly likely that person wouldn't get the diagnosis if assessed anyway
This, with large jingly jangly multicoloured seasonal bells on!
To receive a diagnosis,
The DSM-5 Manual defines autism spectrum disorder as “persistent difficulties with social communication and social interaction” and “restricted and repetitive patterns of behaviours, activities or interests” (this includes sensory behaviour), present since early childhood, to the extent that these “limit and impair everyday functioning.
Emphasis on the to the extent that these impair everyday functioning
That's also why there's no such thing as 'mild autism' if someone is not impacted to the extent that their traits “limit and impair everyday functioning” then they will not receive a diagnosis.
www.autism.org.uk/advice-and-guidance/topics/diagnosis/diagnostic-criteria/all-audiences
Similarly, a medical diagnosis of autism is not a label, they are for jars of jam and suitcases 
Whatafustercluck sorry to see you're struggling to get a referral for your DD.
You need to make a case as the GP's likely to think if her behaviour is only happening at home and not school, then there's a good chance it's parenting not neurodiversity. This is 'wrongthink' on the GP's part, but again it's how the system works and is something parents have to battle with.
I'd suggest you spend some time and document instances of your DD's 'different to peers behaviour' in different locations like supermarket/shopping centre, the park, childrens party, playdate at home, playdate at host's, group activity, team activity, solo activity, her response to instructions from you and from other adults. The object of this is to outline where and how her behaviour is different to her peers and to provide enough evidence that an assessment is needed.
All behaviour is communication, so maybe read as much as you can about likely triggers, especially sensory ones. This is a great place to start.
www.falkirk.gov.uk/services/social-care/disabilities/docs/young-people/Making%20Sense%20of%20Sensory%20Behaviour.pdf?
Also, you may find that making your communication to her more direct has a better effect.
In your example above, your wording to her about "we could try again when she's calmer." is vague. Maybe she needs more structure with instructions.
Have you tried now, next and then instructions with her, so she knows what's happening right now, what will happen immediately and what will happen after that.
Often neurodiverse kids are 'lost' because their need for order is missing. (School can be very orderly hence showing no signs there)
Instead of saying something like "we could try again when you're calmer" which isn't specific, try reframing that to Now you're upset, next we'll have a hug (or whatever calming thing she responds to) and then we'll put your socks on.
It takes a lot of time and a lot of different parenting skills and economy with words, but it's worthwhile.
I'd also suggest if it's possible for you, a private diagnosis for your DD, carried out by a team who also diagnose for the NHS, that way the NHS/LA cannot argue that the diagnosis isn't valid.
In the interim, a private assessment by a sensory paediatric OT could also provide evidence that your GP would take notice of for further referral if you want to pursue a NHS assessment.
Ross Greene's Lives in the Balance and The Explosive Child are worth a look for reasons why your child lashes out and interventions to help.
Free printable resources for tasks like now, next and then instructions and much, much more are available from www.twinkl.co.uk/