with/without + pronoun (me vs. my) + gerund-participial phrase

The Cambridge Grammar of the English Language (Page 461) has this section:

(f) Subject of clausal complement of with/without

Pronouns in this position normally appear in accusative case:

[16] i We set off again, the Rover going precariously slowly in very low gear up hills, with me staying on its tail in case it petered out altogether.

ii With me out of the way, there would be no one to curb his excesses.

Note that {this is one place where a gerund-participial in complement function cannot take a genitive subject}, but unlike the construction dealt with in (b) above the accusative is not here an informal alternant to a nominative.

I think CGEL is saying that using 'my' instead of 'me' in [16i] is wrong:

(1) *We set off again, the Rover going precariously slowly in very low gear up hills, with my staying on its tail in case it petered out altogether.

More importantly, it seems to me that the bracketed portion of CGEL is making a blanket statement that the subject of clausal complement of with/without cannot take a genitive form (possessive form). That is, I think "this" in CGEL's bracketed portion refers to the entire case of (f) Subject of clausal complement of with/without, not just [16i] or [16ii]

I for one wouldn't use 'my' in [16i], but that's just me. Theoretically, I know that you can use both 'my' and 'me' as the subject of V-ing, as shown in this question 'When is a gerund supposed to be preceded by a possessive adjective/determiner?'.

In that question, there's this sentence having 'without':

(2) Most of the members paid their dues without my asking them.

And it has been said in the answers there that 'my' as well as 'me' is possible.

Q1. Is it correct to say example (1) is ungrammatical (with 'my' instead of 'me')?

Q2. If indeed CGEL is making a blanket statement in the bracketed portion, how do you distinguish this blanket statement from example (2)?

I think "with me -ing" and "without me/my -ing" are at least two different constructions. "My" is not possible in the first construction.

Construction 1: [accusative NP] [predicative complement]

[16ii] shows that "with" can be followed by an accusative-case pronoun and then by something that could function as a predicative complement (like the prepositional phrase "out of the way"). That construction would be expected to allow a gerund-participle, since that can serve as a predicative complement. (I think that is the correct analysis; anyway, a gerund-participle can serve as the complement of the copular auxiliary verb be. I don't know whether the way I use "predicative complement" in this post is completely standard terminology, but that's what I mean.)

Just as we can't say *"With my out of the way", this structure wouldn't permit the use of "with my -ing". So any "with(out) my -ing" forms that are grammatical must have a different structure.

Gerund-participles can participate in a distinct construction (or constructions): [genitive NP] [gerund phrase] or [accusative NP] [gerund phrase]

I think that in "without my -ing", the "gerund clause" "my -ing" acts like a noun phrase; the structure would be analogous to "without (my) assistance" or "without my cooperation". Therefore, the meaning is something like "without [the event/action of] me/my asking them".

I don't think "my" works in [16i], because I don't think that sentence is intended to express "We set off again, the Rover going precariously slowly in very low gear up hills, with [the event/action of] my staying on its tail in case it petered out altogether."

As usual, "my -ing" could be replaced by "me -ing" in this context. But I don't think "me -ing" in this context would have the same grammatical structure as in "with me staying on its tail in case it petered out altogether".

"me -ing" can show up in more than one construction

For comparison, "You won't catch me doing that" likewise cannot be replaced with "You won't catch my doing that". From a traditional standpoint, I think the distinction between situations where "me -ing" can be replaced by "my -ing" and situations where it can't could be summed up as "you can replace 'me' with 'my' when the -ing verb is a gerund, but not when the -ing verb is a present participle". From a standpoint that rejects the gerund/participle dichotomy, I'm not sure what the best way of describing the difference is, but I do think there is a real difference between various constructions involving the sequence "me + -ing verb".