j.c.: Additionally, I think you’ll find that science fictions writers have successfully dealt with morphing characters for decades.
Well, with varying degrees of success: Ursula K. LeGuin wrote The Left Hand of Darkness about a society where people routinely changed their biological gender, and used male pronouns to refer to all of them; but she has since commented that she is not satisfied with the effect (and I must say I found it pretty automatic to think of all these “he” and “him” referents as male by default, even when I consciously tried to remember they morphed). LeGuin wrote a short story “Winter’s King” about the same society where she kept male titles (“king”, etc.) but used female pronouns, IIRC.
However, I think what the OP was talking about in the case of a “fictional person whose sex may vary” was not sci-fi gender morphing, but plain old gender indeterminacy: i.e., the indefinite “one” whose gender is not known. If you don’t know who’s reading your book, you can’t really say “The reader who likes his fiction light and cheerful…” or “The reader who likes her fiction light and cheerful…” without sometimes being inaccurate.
Some writers still follow the old fashion of always using male pronouns in gender-indefinite cases, and many readers find it a little obtrusively old-fashioned or even slightly sexist, but probably nobody really minds very much. Some writers follow the new fashion of always using female pronouns in gender-indefinite cases, and many readers find it a little obtrusively PC, but probably nobody really minds very much.
Some writers try to use either male or female pronouns more or less at random in such cases; some writers use plural pronouns with singular verbs and don’t worry about the grammatical mismatch; some writers use neologisms like “hse” or “hirs”; some writers just scrupulously avoid any constructions that require a singular gender-indefinite pronoun. (The children’s author E. Nesbit, I recall, always used the neuter pronoun: “Everybody put on its own warm clothes…”.)
I don’t really think it matters that much which you choose. But I approve of the current chaotic situation where there are lots of different customs co-existing, because I find that it has the effect of making the semi-conscious “gender reaction” (“why do you say ‘he’?” “why do you say ‘she’?”) easier to ignore. And that’s the whole point of gender-indeterminate pronouns in the first place: they’re supposed to let you think of “a person” without making any subliminal assumptions about his/her/their/its/hirs/one’s gender.