Nice ditty.
What reason do you suggest why “his or her” would be preferable to “their” in this context?
Regional dialect, fluidity of language, variety - even habit.
“It’s grammatically incorrect” argument doesn’t hold much water
Oh, I do respectfully disagree with that, especially when you cite medieval English but reference an American language dictionary as your source.
I could just as viably give "his or hers" as equally valid as "theirs", because it is. We're not newspaper headline writers, nobody penalises us if we use a few more characters for any reason. And you could switch back and forth between them both for variety.