Although I wasn't totally unsatisfied with any of the available options (in fact, I invented my own - see the table's last row), I originally settled for MacKay's approach. Soon after I submitted the first version of this post, however, I discovered that there once were GNPs native to the English language (or, at least the version spoken between the High and Late Middle Ages - i.e., Middle English). These included the pronoun "ou" (as in ou smells funny, in place of he smells funny). What I find especially fascinating is that "ou" is also used in Farsi, another Indo-European language.
Anyway, I wish to declare that I have joined the movement (actually, I doubt it's a movement - I'm guessing that 99.99% of the English-speaking world wouldn't give a rat's ass about gender-neutral pronouns) to revive the use of "ou"!
We'll see how long this lasts...