I'd say the headteacher is not looking at this from a person centred point of view (ie DD's), but rather from a staff management point of view!
DD had two TAs in a job share for 2 years. I think one did Monday to Wednesday midday; and the other did Wednesday midday to Friday. DD was fine with that. Next year, in their infinite wisdom, they decided to give her a classroom TA every lesson - so a different one for each subject. They did not understand her particular needs - so for instance, one was very "loud"; when DD can't cope with "loud"! She needs quiet and calm (for a comprehension point of view, apart from anything - she can't keep up with people, who talk fast). DD needs routine and predictability. She took to running out of the classroom, because she could not cope, and ended up refusing to go back regularly.
We have found DD does get attached to people; but that is only human - however, if they said goodbye to her at the end of the year, she could cope with it, even though she was upset. People disappearing for no reason, as far as she knew, was worse!
I suspect your DD needs routine and predictability too - she could cope with a few TAs in a regular timetable (using a visual timetable for her, so she knows on Monday she gets X, Tuesday Y, etc); and they could get used to her and how she thinks. I would have thought if the TA changes every few weeks (if you are on the 5th change already), they are just getting to know what makes her tick, and they get moved on?
I was talking about staff changes with DD's clinical psychologist. He said we all need external cues like clocks, routines, etc to keep us orientated. With frequent, unpredictable staff changes, children and young people get disorientated. Solitary confinement is an extreme punishment, because people get very disorientated with it.
I wouldn't take any notice of LA policies like they never do......They are not supposed to adopt blanket policies; they have to consider the needs of the individual - I'd be asking for two or three TAs trained and experienced in ASD, in a job share, so at least DD has some routine and predictability.