DD's SN was picked up at pre school and diagnosed by a SALT just before she went into reception. In reception NHS SALT gave advice to class teacher and TAs re SM. From Y1 to Y4 DD had weekly SALT in school from an independent therapist whose services the school 'bought in.'
SALT worked with DD to get her speaking in class (DD only spoke in the playground to friends, and occasionally staff). DD managed to speak in a small group outside the class supported by the SALT, and then in a group led by a TA. But her speech never 'generalised' to the classroom.
I did ask for DD to be referred to a paediatrician when she was in reception as my DS2 had just been diagnosed with HFA - but the paed only took a history then dicharged her to SALT services.
Behaviour wise anxiety has always been the major trigger for DD - refusing to walk in unfamiliar places, not coping with social occasions, difficulty separating from me. On the positive side DD has managed to make and maintain friendships, though parties often meant sensory overload and tears.
In Y2 DD's anxiety ramped up to the extent that she wasn't always able to do the work in class - she just 'froze'. DD was referred to CAMHS and back to the paed who put her on the list for ASD asessment. DD was offered psychotherapy by CAMHS to take the form of play therapy. We weren't keen because we thought even if it helped DD in the session, it wouldn't help her in school as SM is obviously situation specific.
DD wouldn't go into the sessions without me and wouldn't, or more accurately couldn't, engage with the therapist. After about 5 sessions it was decided that DD wasn't 'compliant' with the therapy and it was stopped.
By this time DD was in Y4 and her anxiety meant that her attendance at school was suffering. By the middle of Y4 she was only in school for an hour a day in a room away from class with a TA and me. In the May the GP signed her off from school, whilst we were waiting to see a doctor at CAMHS. She was prescribed medication for anxiety.
During DD's Y4 I applied for a Statement (as it was then) and it became obvious DD wasn't going to manage to re integrate into her mainstream primary. We found a very small specialist primary and it was named on her statement. The combination of a small supportive primary with SEN experience and almost certainly the medication too, meant that DD settled really well into her new school - and started talking in class and to teachers for the first time! In y5 and y6 she managed residential trips with the school!
It sounds as if you already have lots of professionals on board for your DD - I think a SALT with experience of managing children with SM is crucial.