Are you in the UK?
How old is your dd?
High functioning Autism runs in my family and I’ve worked in special needs schools in the past and my husband is a specialist youth worker supporting autistic children
This is a good enough reason to go to your GP and ask him to copy in your paed. Autism is believed to be hereditary and autism presents differently in girls.
Have you done any online tests, MChat-R for littlies, if she's way over age for that, have a think back in time and try to fill it in yourself anyway, for your own understanding of things she did when she was young. One part of the diagnostics (depends which ones you have) which you may or may not be given is questions about the person aged 4.
Find an online AQ test for teens and over. Read as much as you can about presentation of autism in girls and women.
I’m just scared in general that if I seek help and it’s confirmed that the label could do more harm than good (what her teacher hinted at)
It's not a label, it's a medical diagnosis like any other medical condition. Would the teacher dissuade you from seeking a diagnosis of any other medical condition?
Perhaps you'd like to enlighten the teacher, suggest she's sent on a disability awareness course, one that has emphasis on hidden disabilities.
Teachers in general know nothing about presentations of SN.
Your daughter will either have autism or she will not. A diagnosis after comprehensive assessment by medical professionals will open the door to school having to spend some money put some support into place for her and possibly exam concessions etc. Support in FE and later work.
How is she emotionally? Many kids who have SN are very emotionally immature, rough guide is about two thirds of their physical age which contributes greatly to their vulnerability. This also needs to be taken into account by yourselves and school.
The gold standard for assessment is the Lorna Wing Centre, it's expensive but although it's private, it has credence with the NHS.
Some local authorities don't accept private diagnoses, they insist on NHS but that can take around 2 years depending on whereabouts you live.
Get the ball rolling as soon as you possibly can.