They are supposed to comply with the law, and if your statement says 'experienced with ASD' then they must be experienced. The trouble is that your statement may not say how experienced. If they are a TA who started last year in a class with one ASD child, then they have 'experience'.
It is pretty rare these days for a teacher to qualify without one of the pupils that they come across having ASD.
The only way to enforce the terms of the statement of needs through the court is via JR and by the time you had filled in the form your 3 weeks would be up.
So I think you should work on the quickest way to get to what you want.
The first question is what experience of ASD the person has.
If you would like it to be greater experience, then suggest that you do some research to find a temp who would like to do the job and who you are happier with. Obviously, (you can say) if you can get a regular temp it minimises the disruption for the school staff, other pupils in the class and DC. Sell it as a win-win-win.
They get cover teachers from somewhere, so ask for that source and see if they have useful TAs listed.