Form of Participle Clause Subject: 'his' being or 'him' being

The opposition leader's bid for power failed to gain traction, and his claim to the presidency is rooted in his / him being the head of the current legislature.

The pronoun subject of gerund-participial clauses functioning as a complement can be either genitive case or accusative case. The only difference is one of style, with the genitive case being characteristic of fairly formal style. Syntactically, the pronoun "his / him" is subject and "being the head of the current legislature is a verb phrase functioning as predicate.

Note that when a gerund-participial clause is in adjunct function, genitive subjects are not permitted at all: the choice is between nominative and accusative:

She sought advice from Ed, he / him being the most experienced of her colleagues.

Both his being and him being are valid in restricted contexts. Since English grammar is largely analytical, the judgement depends on the broader context, specifically the preceeding verb. Arguably, the claim is rooted in something of his, not in him. Conversely, had they opted for the verbal phrase come from, I think came from him would be more readily acceptable and being... could then be parsed as a present participle. Instead we have to read his being as a nominal gerund, which I find less optimal, given that being is not a substantive, but a process.

It's not a particularly fine example of formal register, overall, and appears more like out of a polemic letter written in the heat of agony, as it's meaning is barely understandable. wow, a real garbage fire.