Be strong and go with your instincts. My DD is just 4 and over a year into the system for diagnsis now. My eldest DS is 11 and has an Autism diagnosis. My HV said when he was little and I was concerned 'it can't be ASD he's lovely'. She's right he is lovely but its deffinately ASD too. He has imagination, so do many ASD children we know, which was another excuse thrown at me.
Have you done any of the online assessments to ask to put on your DD's file?
Did speach and language do a full assessment? With DD's assessment she refused to do the dressing up, had poor eye contact, couldn't recognise any of the emotional images (still a bit young anyway for this).
It was enough, with also having a diagnosed brother, to get an in nursery assessment.
What do you want from diagnosis? There are lots of reasons people pursue early diagnosis ranging from not wanting to be labeled as a bad parent, to needing help managing bedtime routine, needing help with physical development challenges, needing help with eating, litteral language issues so speach and language support.
A diagnosis is not the end of things either. Its just the start of a whole heep of new challenges to get the right support packages in place. At the end of the day behaviour is behaviour and sadly diagnosis or not you're going to need to be the one to pick your battles and work through which things to prioritise to work on managing.
Don't give up. Just keep gently plugging away until you're heard. Keep a diary of extremes of behaviour and daily challenges/ routines. Does your DD have good motor skills i.e. Hold a knife and fork correctly, jump, hop, catch a ball? When you're with similar aged children make notes of behavioual and reaction differences to show peer group variation. Build your own paperwork trail and take it back to the GP in 4-6 months asking for refferals for the areas you want assistance with.
DS got into a special Autism unit nearly two years ago. We moved areas partially in the hope of accessing more support. Its made the world of difference.
Both DD and DS enjoyed big jigsaws at a very young age. Both read early, both had a very good grasp of number at an early age. DD is pre school (starts september) but is pretty confident on her number bonds to 10, and knows various random multiplications because we've been running through them with my nine year old and she picks them up. She can confidently divide and group by single numbers. She can't understand why everyone is not her friend, eat when overstimulated, finds it very challenging to be presented with new foods, likes her food in the same order not touching on the plate(s).