You’re very hung up on whether the form “me’s” is a word. hazel never claimed that “me’s” is a word, and linguistically, it is of no relevance, since it is only a matter of convention that in written English symbols are grouped and separated by spaces to form “words”.
All native speakers, whether educated or not, have a grammar in their heads that allows them to construct sentences in their native language. This grammar has little to do with style guides.
The inconsistency is as follows:
Usually, we distinguish between
A) Ann and Betty’s cats
B) Ann’s and Betty’s cats
where A) refers to cats owned by Ann and Betty, and B) refers to cats owned by Ann and cats owned by Betty.
In this case, your style guide recommends:
C) Bill’s and my car
Now, suppose there is more than one car, the style guide would prescribe:
C1) Bill’s and my cars
We don’t know whether it is the cars belonging to Bill and me (shared ownership), or the car(s) belonging to Bill and the car(s) belonging to me.
Therefore, to avoid sounding silly, we have suppressed a useful distinction and re-introduced ambiguity.
Finally, when you add apostrophe-s to a noun phrase to make a possessive form, it is indeed the whole noun phrase whose case is changed. Note the difference between
A) the ((big boat)'s) wheel
B) the big (boat’s) wheel.