I don't advise taking the child to a psychologist without consent of the other party, or at least trying to reach agreement about it. The reason I say this is because it's a significant medical decision, particularly for a child of such a young age, and not including the other party shows they're not valued as a parent and that there is no intention to co-parent amicably for the benefit of the child, which the court frowns upon. Further, a family law psychologist will have little bearing on proceedings unless they can also speak to the mother because they will otherwise have an opinion considered by the court to be biased.
The father would be better asking the court to order a family report to assess the child's well-being and relationship with both parties. A family report involves interviewing both parents and observing the child in interactions with the parties. Parents rarely see the full scope of their child's relationship with both parties because kids will often say what they feel pleases the parent most. For example, a mother who denigrates dad will often hear from the child about how the child is scared of him or doesn't want to see him and uses that to justify withholding the child, while the reality is that the child probably enjoys time with dad very much, but he/she is more or less agreeing with mum to mum's face because that's what the child thinks will make mum happy. A family report breaks through that dynamic and gets to the core of what the child is experiencing in each household.
I hope this helps.