My daughter had a psych eval at age 2 and she's had several since.
There have been different suggestions depending on her age, and different strategies suggested, and different therapies or help. However, there has been one consistent thing throughout all the evals she has had, which has been invaluable, and which I'd like to suggest that you consider.
And that is: we, as parents, learned so much. When she had her first eval, we already had a son who was healthy and thriving and well-behaved and happy. We figured that as educated, loving, stable parents who weren't first-time parents, or completely inexperienced, or neglecting our kids, we pretty much knew the basics of parenting. We assumed that the same basics that applied to our son would also apply to our daughter. Sure, she had some issues that our son didn't need to deal with, but basic eating, teaching, sleeping, etc should apply to both kids, right?
How wrong we were! The first psychiatrist kindly acknowledged that our son was doing very well, and that we were doing everything right, but that sometimes, for some children, the rules just have to be changed, and what's right for one kid, or standard for most kids, just won't work in some situations. Standard parenting techniques that worked with our son just wouldn't help our daughter. We had to be receptive to the education that the psychiatrist was willing to give us. Our daughter's issues meant that we had to learn new ways of interacting with her, teaching her, helping her.
And we took his advice to heart. I really think we got more insight, advice, and understanding out of these appointments than our daughter did. Of course, she was young, but even today, psychiatrists have helped us gain a better understanding of her (she has medical issues, and anxiety and depression).
So, the testing is pretty painless, and at such a young age, the kids often see it as play, or fun. But the help that parents can get, if they're receptive, and willing to learn, can be invaluable. If you approach a psych eval as an opportunity to learn about your child, and an opportunity to fine-tune your parenting techniques, and an opportunity to gain some invaluable insight into the kind of child you have, and a way to help her have the best possible life with the best possible parents, then I suggest that an eval can be a great thing. She may not be shy, or she may not be what you think or already know. You might be surprised, and if you tell the psych that you're willing to learn, you might find that you have a great partner in the doctor, who's willing to help you.