Yes, indeed- it does sound like those strategies need to be devised with a specialist who has assessed her in detail.
I'm afraid I don't know of anyone who will do this privately (hopefully more posters who do will post), but you can ask for a referral for a Clinical Psychologist assessment via your GP. It sounds as if a diagnosis will help her access support.
I have to say, I would be concerned by a school which did not see a child's social development as part of its remit. Your Dd is very young, and this development is part of her academic development. Does the school follow the EYFS?
IME, many children with challenging behaviour suffer from low self esteem/confidence, and the withdrawal is another side of the same coin IYSWIM. Strategies to help her feel confident speaking out will vary in their efficacy depending on what exactly it is holding her back, as this will vary from child to child. My DD (5) has only now started to make verbal contributions in class, and has been at the school for a year. For her, it was a case of working out and reliably predicting the reactions of other children. For your DD, it is likely to be something else.
It may be worth having a look at Circle of Friends as a strategy to use at school. IME, it is very effective.
Again, I'm not sure how it works in the private system, but I would be asking school to refer your DD to the Educational Psychologist for an observation and an assessment. You may have to fund and arrange this yourself; if this is the case, make sure school will facilitate this.
Good luck 