It was a couple of years ago - from memory I think I brought up the subject verbally (casually and briefly) and gave her the book to read, saying something like 'see if you think the girl in the book sounds like you when you find it difficult to speak to people'. I then left her to read it.
She did read it all, her response was pretty much 'meh - it only sounds a bit like me', but she seems to have taken on board the term selective mutism as she uses it to (often crossly) explain why she doesn't want to do something as it involves speaking. However I have heard of other children loving the book and identifying with it immediately, so it seems to depend a lot on the child.
I asked her today whether she had found the book helpful and she said 'not really, it tells you what it's like to have selective mutism, but I already knew that, didn't I!' So honestly, I don't know - on balance I think it was useful for her on some level, but possibly more helpful for giving to teachers / relatives. I guess the key is to let the child know you understand how difficult it is, but also support them in combating the SM even when they have no interest in combating it. It's very hard and I certainly haven't cracked it.
DD has got better at speaking in school but isn't 'cured'. We've had a few attempts at sliding in, DD was resistant to it when it involved going in at 'quiet times' as she felt even more self-conscious, so that didn't work. I think things like going in to help with class activities such as reading has generally been helpful, but it hasn't been a miracle cure. It sounds like your school is doing the right thing by not applying pressure etc. Do they do any small group work targeted at improving social skills? Those can be very good when done well.
Have you joined the SMIRA group on Facebook? It's very active and helpful if you ask questions, and has a lot of written resources.