As you imply, he was never a Hamilton by birth and only changed his name, as he also changed his religion, in order to marry an advantageous wife, to whom he was unfaithful. Nevertheless, my own view is that by the time of his death we could allow him to have become a Hamilton.
You may also be concerned that the Wikitree profile has oddities. The alleged place of birth is fictitious, as he was not a Hamilton and was probably born in a Douglas possession. And the place of marriage is perfectly well known, being at Corstorphine in Midlothian.