There's a simple reason why subject pronouns like him and me should be objective here.
There are, as noted, any number of different ways to report the same proposition.
But there is a very limited number of possible complement clause types in English.
There are only four of these types of clause [bracketed below]:
two finite clause types, requiring a nominative subject and a verb in past or present tense.
- that clauses : I think [(that) he has left].
- wh- clauses : I know [what he wants].
two non-finite clause types, each requiring a non-nominative subject and a non-tensed verb.
- infinitive clauses : I wanted [(for) him to leave]
- gerund clauses : She deplored [him/his leaving so soon]
Non-finite complement clauses often lack a subject, if it's indefinite, like the subject of leaving in
- [Leaving immediately] could be misinterpreted.
or if it's predictable by some syntactic rule, like the way we identify the subjects of leave and want
- She wants [to leave soon].
When a non-finite subject is omitted, it leaves only a verb phrase, like those bracketed above.
These are sometimes called "participles" or "participial phrases", but they're really just clauses that have lost their subject somehow, because the subject can always be determined.