So Prigozhin just forgot about his family before starting the fight? Or didn't think anyone would think to threaten his family? Seems very unlikely to me.
But we're talking about brinkmanship in a decades-old personal relationship between two individuals (Prig. & Put.) whose senses of context and honour patently are warped by ordinary standards, as evidenced by their past and ongoing actions.
It seems not at all far fetched to me that, in such a strange relationship, each others personal family would be seen as sacrosanct... up to a point.