possessive - Adverbs modifying nouns?

1. What this question is about

It is about cases where an adverb apparently modifies a word of a type that adverbs aren't supposed to be able to modify, like nouns and personal pronouns. It is very much related to this post. Prototypical examples I have in mind:

The work is mostly Kim's.
The work is mostly mine.
The victory was almost Kim's.
The victory was almost hers.

2. The actual question

It seems to me that there are three possible ways to analyze the examples above, none of which seem to me to be satisfactory. Question: which of these, if any, is the best way to analyze the examples above? I would appreciate it if the answer was backed by scholarly sources at the level of Quirk, Greenbaum, Leech, and Svartvik and Huddleston and Pullum (as opposed to school or ESL grammars).
Possible analyses:

i. The NP answer: Although, true, Kim's is a noun, and a nominal, in these examples it is actually the full noun phrase (NP); similarly for mine and hers. And while adverbs "do not occur as attributive modifiers within a nominal, ... many can occur as external modifier with an NP as head" (see Sec. 4.1, below). So there is no problem: NPs can be modified by adverbs.

ii. The adjective answer: Kim's may be a noun, but here it functions as an adjective; similarly, mine and hers may be pronouns, but here they function as adjectives. So there is no problem: adjectives (and, by extension, words that function like adjectives) can be modified by adverbs. (A variant of this analysis says that all these nouns and pronouns actually are adjectives in these examples.)

iii. The verb answer: In these examples, the adverbs mostly and almost actually modify the verbs is and was. So there is no problem: verbs can be modified by adverbs.

3. Why I don't like the answers 1-3

3.1 The NP answer

This was the accepted answer to this question; so why don't I like it?

I don't like it because it seems to open a Pandora's box. After all, any noun can be the sole constituent of an NP. This analysis would imply a vast number of circumstances under which nouns may be modified by adverbs---potentially, all circumstances in which a noun is the sole constituent of an NP. What then remains of our analysis of adverbs as those words one of the key characteristics of which is that they don't modify nouns?

Moreover, such an analysis does not seem to capture the extremely limited range of circumstances under which adverbs (seem to) modify nouns and pronouns. For example, surely, the following sentence is not allowed in Standard English:
* Mostly John did the work.
But, on this analysis, why not? After all, John is clearly NP. It is even clear what the sentence means; moreover, I think the sentence is actually grammatical in some dialects (e.g. in Indian English)---but it definitely isn't grammatical in Standard English.

3.2 The adjective answer

My guess is that this is the preferred answer of traditional school grammars, because such grammars say that in the phrase the Clinton administration, the word Clinton is an adjective (or that it is "used as an adjective"). Why don't I like it? Because of what Huddleston and Pullum have to say about it:

Traditional school grammar (though not scholarly traditional grammar)
tends to analyse the underlined nouns here as adjectives---or to say
that they are 'nouns used as adjectives'. From our perspective, this
latter formulation represents a confusion between categories and
functions: they are not nouns used as adjectives, but nouns used as
attributive modifiers. Apart from pronouns, just about any noun can
appear in this function---including proper nouns, as in the London,
Clinton, and Caroline examples [in the book there are many other
examples, and the nouns in question, e.g. Clinton, appear
underlined]. These words can all appear as head of an NP in subject or
object function, where they are uncontroversially nouns; to analyse
them as adjectives when they are functioning attributively would make
the adjective category far too heterogeneous, and require an
unwarranted and massive overlap between the adjective and noun
categories. (p. 537)

It would seem to me that the same remarks apply to the case of nouns and pronouns in the genitive (see Sec. 4.2, below).

3.3 The verb answer

I'm not aware of any sources that advocate this answer, but it seems an obvious possibility to consider. However, either one of the following two reasons is probably enough to see why no one is advocating this:
i. adverbs normally come before the verbs they modify (e.g. I want to mostly eat, with some drinking mixed in.);
ii. it would never occur to anyone to analyze It is mostly red as anything other than red being modified by mostly.

In conclusion, I don't much like any of the proposed answers. Is there another possible answer? Or is one of the ones above in fact the correct one, despite the objections I stated?

4. Appendix: Some grammatical background

4.1 Adverbs

According to Huddleston and Pullum,

Adverbs do not occur as attributive modifiers within a nominal, but
many can occur as external modifier with an NP as head. Almost the
whole book, for example, has the NP the whole book as head, and may
be contrasted with * She congratulated him on his [almost success],
where it is inadmissibly functioning as modifier of the noun
success. (p. 563)

It may be helpful here to recall the distinction between a nominal and a noun phrase:

A phrase consisting of a noun and the constituents that go with it
most closely is a nominal [Nom]; a nominal plus a determinative makes a
noun phrase [NP]... (p. 22)

Example (p. 23):

this clear case of dedication to duty is an NP, whereas

clear case of dedication to duty (note the missing determinative) is a Nom.

It is helpful to note that, if clear is dropped from the above example, then

almost a case of dedication to duty

is grammatical (if awkward), but

* almost case of dedication to duty

isn't grammatical.

4.2 Nouns and pronouns

According to the source I cited, the genitive form of a noun is still a noun:

Nouns prototypically inflect for number (singular vs plural) and for case (plain vs genitive) Examples: dog-dog's; dogs-dogs'; child-child's; children-children's (p. 326)

Likewise, pronouns in the genitive remain pronouns (p. 327):

nominative     dependent     independent

                            genitive           genitive

          I                     my                 mine

        she                   her                 hers

4.3 The relevant genitive construction

The book lists six types of genitive construction (p. 467). Of relevance to this question is type v, the predicative genitive:

v   All this is Kim's.     [v: predicative genitive]

Details are provided on p. 469:

In [41V] the genitive marks the relation between Kim and the predicand all this, a relation like that expressible by belong + to. Kim's here is thus not part of some matrix NP, as it is in Types 1 and iii-iv: here, then, the predicative complement function is realised directly by a genitive NP. Genitive predicative complements are usually subjective, as in this example, but they can also be objective: Let's call it Kim's; I regard it as Kim's.