In addition to mtwde's answer:
Indeed, a child psychologist in the sense of a child who is a psychologist is not a thing you meet so often in reality, or expressed the other way round: even if you come to the conclusion that every child is also a psychologist of some kind [yes, pun more or less intended], this is not a standard topic of communication, and therefore there is no special term for a pycholochild.
People coin terms for things they address frequently. For things that are no topic of communication, there are no words, usually. (Interesting question: Is there a word in any human language relating to a thing about which nobody speaks?)
However, if you take a little bit a broader perspective, you may find suitable terms. Mtwde mentioned Junior-Psychologe, which is an option, but it is not perfect because the first thing that would come to your mind when hearing this is something like an assistant psychologist, a psychologist in training or so, so usually a person in their late twenties or early thirties.
However, for children plus professions there is a tradition of calling this
Der kleine Zauberer
Der kleine Chemiker
Der kleine Zimmermann
Der kleine Radiologe 1
Der kleine Psychologe.
You would usually find such terms on books and on experimental kits for children for topics like chemistry, engineering, physics, biology, illusionism etc. The term can be used both for the book or experimental kit (cardboard box with variuos materials) and for the child playing or experimenting.
As far as I know there is no experimental kit "Der kleine Psychologe" but if you think this is a market niche: go ahead and offer one! What would you include? I suggest: a desktop calender; a couch; a coffee table an three armchairs; a note pad and a pen; Rohrschach test cards, a potted plant, and a set of tiny dolls for familiy constellations.
1 This is only partly a joke. There was an experimental kit available in the 1950 in the US for exploring radioactivity (Gilbert U-238 Atomic Energy Laboratory). The kit contained materials such as uranium ore plus various instruments such as a Geiger counter and a cloud chamber. For some reason, this product line has been discontinued... It was also rather expensive; if the information I found is correct, it was something equivalent to today's 500 dollars. Nuclear lab kits only for the better-off child, it seems!