Hi Philip

You have it right. Where a pronoun is subject of a clausal complement, it normally appears in the accusative case, i.e. "him", not "he".

However, some examples (probably mainly literary ones) like yours with a third-person pronoun seem acceptable, whereas others don't:

*"We set off again, with he following closely behind".
*"With they out of the way, I hate to think what might happen".

In those two examples, nominative he/they sounds absurdly formal to the point of being unacceptable to most speakers.

I would say stick with the accusative and you can't go wrong.

PaulM