I can only find one other proper Dornishman joke, of the genre you refer to, although the punchline is not revealed. From A Storm of Swords, "Tyrion V" (the same chapter your joke was from):
"You want to die with a breast in hand, you said." Tyrion cantered on ahead to where the ferry barges waited on the south bank of the Blackwater. He had suffered all he meant to suffer of what passed for Dornish wit. Father should have sent Joffrey after all. He could have asked Prince Oberyn if he knew how a Dornishman differed from a cowflop. That made him grin despite himself. He would have to make a point of being on hand when the Red Viper was presented to the king.
And this is not actually a joke, but it refers the joke setup you mention in your question, and it's also from the same chapter:
A Martell sun, but ten years too young, Tyrion thought as he reined up, too fit as well, and far too fierce. He knew what he must deal with by then. How many Dornishmen does it take to start a war? he asked himself. Only one. Yet he had no choice but to smile.
I guess this can count as a joke; A Clash of Kings, "Tyrion IV":
"This secret is your lord father's sworn man."
"My father would be the first to tell you that fifty thousand Dornishmen are worth one rabid dog."
This is not exactly a joke, but a humorous stereotype from A Feast for Crows, "Prologue":
Leo's eyes were hazel, bright with wine and malice. "Your mother was a monkey from the Summer Isles. The Dornish will fuck anything with a hole between its legs. Meaning no offense. You may be brown as a nut, but at least you bathe. Unlike our spotted pig boy." He waved a hand toward Pate.
A Game of Thrones, "Eddard IV", mentions a similar humorous stereotypical remark:
"The Crown is more than six million gold pieces in debt, Lord Stark. The Lannisters are the biggest part of it, but we have also borrowed from Lord Tyrell, the Iron Bank of Braavos, and several Tyroshi trading cartels. Of late I've had to turn to the Faith. The High Septon haggles worse than a Dornish fishmonger."
There's also "The Dornishman's Wife" from A Storm of Swords, "Jon I", which is a humorous song about how Dornish women are so fair they're worth getting killed for.
Like you say, there are many other Dornish stereotypes mentioned in the books, like "men said it was the food that made Dornishmen so hot-tempered and their women so wild and wanton", but those aren't jokes or meant to be humorous.
As a bonus, although you asked about the books: We do have another joke in the same genre in the first Game of Thrones episode of the fourth season, "Two Swords" (video), although we don't learn the punchline here either:
Bronn: "How many Dornishmen does it take to fuck a goat?"
Tyrion: "Please don't."