Is it permissible for a mother to name her son after herself instead of his father's name, especially if the father is well-known and refused to provide proof upon his mother's request, then later returned to ask for the son to be registered under his name after the mother had already registered him under her name in another country that permits it?

The child is attributed to his father if he is known, based on the Almighty's saying: "Call them by [the names of] their fathers; it is more just in the sight of Allah." If a legitimate marriage between you is established, then paternity is affirmed for the husband unless he denies it through li'an [oath of condemnation], for "the child is for the [owner of the] bed." The father erred by obeying his mother in not affirming the paternity of his son. And whoever denies his child while seeing him, Allah will veil Himself from him. If the father does not cooperate in affirming paternity, there is no harm in attributing the child to his mother in documents while always calling him by his father. Indeed, some of the Salaf were attributed to their mothers even though their fathers were known.