We had a similar situation, albeit the child concerned was much older, ten years old and the circumstances were more 'matured', shall we say.
My Argentine lady had a particularly acrimonious split from her long term partner (they were never married) and when I arrived on the scene six months later, unaware of the nitty gritty, it was almost open warfare.
Later, when we were planning on visiting the UK as a family, the ex demanded an exact itinery of our movements in the UK, including addresses and phone numbers which he could then track on Google maps, before giving written permission for his son to travel with us. I recoiled at this suggestion, but my lady buckled under the pressure and provided it.
He almost immediately refused permission and so we travelled without him, thus denying the child an opportunity to open his eyes to a world outside Argentina.
Our intentions were honest and never in a month of Sundays would we have considered absconding with the child. Furthermore he also refused to agree to renewing the child's passport at that time, although the law has since changed.
Either he had this real fear that we would try to 'kidnap' the child, or he was simply playing his own game. I firmly believe the latter.
Since that time there have been numerous incidents of a similar nature which have merely served to enhance the child's hatred of his father and also a fear of him.
For me, the most abhorrent is the fact that the ex uses his child as a pawn to get back at the the child's mother and he has no qualms in using this form of psychological warfare whatsoever.
For my part, I have suggested to my lady that he deserves only the information necessary, either by law or by means of seeing his son which we have never prevented.
I know this may be veering away from the original OP post, but we know by experience that there are some people who play by their own rules and those alone.
We have learned to read the undercurrent to understand the agenda. Things are more peaceful now as only time can assist in and the child who is now 16, makes his own mind up when and if to see his father.
Sadly, he said to us the other day 'When I'm 18, I will destroy him for what he has done.'
As has been said many times, we reap what we sow.