1. The original sentence with “adverbial clause’ is;
- We appointed Max, because he was much the best qualified of the candidates.
Yes, that is correct. I'm not sure I'd use the comma before "because."
2. Abbreviated form with “nominative absolute construction“ is (here ”being“ as participle);
- We appointed Max, he being much the best qualified of the candidates.
OK.
3. Abbreviated form with "with"-variety of absolute construction as an ACC-ing/Poss-ing construction“ is (here ”being“ as gerund);
- We appointed Max, with him / his being much the best qualified of the candidates.
As we told you in our replies, the absolute construction does not accept a possessive (at least in this position after the main clause, as clarified at the end of this post), so only the form with "him" is correct.
I think this V-ing form would be considered a present participle in traditional grammar.
4. Transformed form with “a complement of a prepositional-phrase construction” (here ”being“ as gerund) is;
- We appointed Max, on account of his being much the best qualified of the candidates.
OK. This shows how the possessive can work in a construction that is not absolute.
5. Abbreviated form with “absolute constructions introduced by ‘with’” (here ”being“ as participle);
- We appointed Max, (with) him being much the best qualified of the candidates.
This is the same as (3).
Now, I think there is a case where "with" can introduce an abridged adverbial clause with a possessive or an accusative. I have searched for "with his being" on COCA and, although most of the occurrences involve prepositional phrases complementizing verbs and adjectives, we can find these interesting examples:
- And now, with his being in the war zone, she still didn't want to hear, though she saw that reading the papers and talking to others was a help to his parents.
Source | FIC: Antioch Review |
Date | 2013 |
Publication information | Spring2013, Vol. 71 Issue 2, p317-326. 10p. |
Title | Quickening: Canada, 1915. |
Author | Ford, Kathleen (AUTHOR) |
- Once again, with his presence, with his being "on track," he wanted to point out the difference between himself and George W. Bush's management during Hurricane Katrina.
- With his being on the road so much, he wanted Dusty to be able to defend herself and her mother.
Source | FIC: Relentless |
Date | 2010 |
Publication information | New York : Leisure ; Enfield : Publishers Group UK [distributor], |
Title | Relentless |
Author | Smith, Bobbi, 1949- |
Unlike in the initial examples from CGEL, the preposition "with" is not optional but required (as is always the case with this type of POSS-ing constructions) and the clause introduced by "with" needs to be placed before the main clause. As for the meaning of the clause, it is similar to that of absolute clauses, being a combination of causal and temporal sense.
Perhaps I should take back what I said at the beginning about the possessive never introducing absolute constructions. I think this works:
- With his being much the best qualified of the candidates, Max was appointed manager.