possessives - 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)?