<p>As used above “but” is a preposition. [him/he] is the object of the preposition. The object of a preposition is the “objective case” of the the pronoun “he” – hence “him”.</p>
<p>Your other examples (because of/ due to/ as a result of) are all prepositions and the same rule applies.</p>
<p>The use of the objective case of the pronoun as the object of the preposition is a basic rule of English grammar.</p>