Both his having and him having are correct, in my view. This is counter-intuitive as we are used to one case being considered correct and one incorrect in what otherwise appears to be a single grammatical construction.
But there is an explanation in the history of English which relates to the perennial argument about whether -ing words are adjectives or nouns, and whether they should be called "present participles" or "gerunds"/"verbal nouns".
Originally there were two words: -ing which was a noun (so his having would be correct) and -and which was an adjective (like French ayant) (so him havand would be correct, in what is sometimes called an "absolute" construction, popular in Latin).
But since both words are now spelt the same we now have two completely different constructions (that would have much the same meaning) looking like variants of the same construction.