Whatever the diagnosis, you will be wanting to sort out provision for DC in the classroom, and ASD kids usually need input from an educational psychologist, a speech and language therapist, and in many cases, but definitely not all, a good sensory Occupational Therapist.
You will have gathered that there ia a performance to go through in some LAs, not all, to get these folk engaged, and to get specific input sorted out, and working.
The NHS OT and SALT can be very wishy washy in their recommendations, so you might feel it is worth costing an assessment and school visit from a good independent EP and SALT. If your DC has sensory issues eg hates noise, loves bright lights, can't stand seams on clothing etc etc, then a good independent sensory OT is a possibility, and they can also cover any issues with holding a pencil or cutlery, and any problems with getting fluent movement when walking etc.
You will have seen the parents who end up in Tribunal - a psychiatrist can be regarded by Tribunal Panel members ie the judges, to be insufficiently qualified to advise on classroom strategies - an Ed Psych is needed.
Hope this helps with the decision