I don't think he's abusive. A bit strict, possibly, but not abusive. Also, that wasn't the reason that Childe's father had him conscripted. According to Childe's official character stories:

• "In the eyes of his father, that third son that he had so worried about had changed for the worse, bringing much uncalled-for havoc to the seaside village of Morepesok."

• "Ultimately, after a huge brawl was pacified with some difficulty and with some near-fatal misses, his father had no choice but to hand his beloved son over for conscription into the Fatui."

• "He hoped that the strict military training of the Fatui could hone his son's temper..."

In other words, young Ajax had returned from the Abyss as the "eye of the storm," chaos revolving around him at every turn, and this sudden change in personality from a timid boy to one who reveled in violence and discord, put his father in a tight spot. He most likely knew of no other possible solution for his son's situation, and seemed to view it as a behavioral issue rather than a supernatural one, since it's implied that Childe never told his parents what happened to him.

However, this portion of his character stories is strange in light of the lore from Ballad of the Fjords as well as Simulanka, and so the question still remains as to how much knowledge his father has, and what hand he's had in shaping Childe's fate. Did he truly know what would happen when he named his son Ajax? Was it his own personal motivation, or the motivation of another individual or larger group? We have no way to say for certain at this point, and can only speculate until we receive more information.