(Then again, the Bible wasn't the place where one'd expect to find jokes in the first place.)

Perhaps not long form narrative jokes, but the bible is actually loaded with humorous word play (puns, double entendre, etc). Unfortunately, pretty much all of it requires a pretty decent understanding of biblical Hebrew. I often wonder if biblical literalists would take such a hard line if they realized the writer was often writing for wordplay as much as for conveying a message.

As I said elsewhere, these sorts of stories (old testament Chuck Norris stories!) aren't about humor. It's "yay Solomon!"

Well, yeah, but... For readers to think “wow, that Solomon guy was so wise!” rather than “that's supposed to be a joke, right?”, the characters would have to have at least some amount of plausibility in their cultural context.

Like the plausibility of these stories?

It's a story about Solomon's wisdom. Whether it actually happened is not really the point.