In essence, you're asking for how possessive pronouns preceding an object of a verb pair with the subjects in a neither...nor construction, especially when that object could pertain to either subject. In other words, what concord is there between a pronoun and its noun phrases in the case of neither...nor? The options break down like so:
Does the possessive pronoun pair with the proximal subject?
Does the possessive pronoun pair with the less proximal subject?
Does the possessive pronoun pair with both subjects, as if they were joined by and?
In order: no, no, and yes.
In the analogous case of verb agreement, which you summarize, several usage guides agree with you, Bryan Garner (Garner's Modern Usage) among them (p. 623), that when there's a difference in number, the more proximal noun phrase determines the verb form. (By the way, neither deal with this issue directly.) Yet I found an interesting note in Pam Peters's The Cambridge Guide to English Usage that might help here. This is under the entry for neither, p. 370:
Questions of agreement also come up when neither is paired up with nor as a correlative conjunction. Again the traditional view was that the following verb [with both subjects in singular] should be singular, and yet research for the Longman Grammar (1999) shows that the use of a plural verb is quite common. In fact singular and plural agreement have slightly different effects.
Compare: N either director nor producer has much experience.
N either director nor producer have much experience.
The singular verb seems to particularize while the plural one generalizes.
A similar observation would apply to the pronouns. Using a singular pronoun particularizes, or puts the focus on the difference between each of their bests. So unless the person means to say they both feel her best or they both feel my best, such a usage doesn't make sense here. A hypothetical usage could be formed out of modifying 2 to reflect this particularization, though the result is awkward:
2a. Neither my friend nor I feel her or my best in the morning.
Using a plural possessive pronoun generalizes - our best - for the sake of simplicity, even when it may be obvious that her best and my best wouldn't be identical. Even so, her best and my best would not usually differ in a crucial way (e.g. her best is manageable pain, and my best is ecstatic joy), so emphasizing the difference matters less:
- Neither my friend nor I feel our best in the morning.
While this usage is rare, I've found at least one example of 3 in a web search:
“The truth is, neither Artie nor I feel our lives rise and fall on hit albums or flop albums,” said Simon. ("Simon and Garfunkel: Class Reunion," Rolling Stone, 18 March 1982)
I have found none of 2a.
Here are a few supplementary examples gleaned from a Corpus of Contemporary American English search, skimmed from relevant results for a collocation search, "nor _nn" (nor NOUN) with an _appge (possessive pronoun) up to four words after. These examples refer to both subjects in a plural possessive pronoun, their, conforming most closely to 3:
Neither Mother nor Father really knew their three little girls. (Helen, by Libby Sommer)
Neither Poppy nor Rose could keep their stories straight on how they'd met. ("All the Way Down" by Kris Saknussemm, in The Antioch Review, 68.4, 2010)
Clearly, neither Carey nor Cannon, her second husband (she was married early in her career to Tommy Mottola, then chief of Sony Music, her record company at the time), is reluctant to advertise their domestic beatitude. (Elysa Gardner, "Mariah finds herself in a happy place: Singer's pleased with her album, movie, and marriage." USA Today, 25 September 2009.)
I did find one exception, but this may prove the pattern:
Neither Kate Kane nor Batwoman ever acknowledges her sexuality, but there are certainly enough clues: (Misha Davenport, "Call me ... A bat for the other team." Chicago Sun-Times, 18 July 2006.)
Kate Kane and Batwoman are two names for the same woman, so her refers to both without difficulty.