How about “honky donkey”? (Though since I pronounce “donkey” as if it’s spelled dunky, this phrase wouldn’t rhyme if I uttered it.)
Hinkey-pinkey: Something an ape uses to wrap packages
ape tape?
gibbon ribbon?
Correct.
Hinkety Pinkety-- Evil Priest
Sinister Minister
Hinkaditty Pinkaditty!
Yeah seriously! Four syllables!
And the clue is a Hink Pink:
bus fuss
Transportation altercation.
Hink-pink: Result of pulling the finger of Homer’s offspring.
Bart fart!
Hinky pinky: vomit oopsie
Kipper flipper?
Hink-pink: Seller of night sticks.
Hinky pinky: 1920s it-girl toilet
Hinky Pinky: buttsecks robber
Yup!
Transportation altercation!!!
Hinkity pinkity: turns your laptop into spaghetti
I was thinking of the latter. Since I try to make mine at least a bit challenging, I wouldn’t incorporate one of the answer words into my clue. 
The ones still available are stumping me, but I’ll add:
Hinkey-pinkey: Sign advertising defunct make of automobile
Oh you think you’re tough do you? 
Packard placard!!
Hinkey Pinkey: twice the disaster
Double trouble, or in German, Unglück zurück (bad luck is back)!
I repeat: hinky pinky: 1920s “It Girl” toilet.
squire fry-er?
Flapper Crapper
Hinkey-Pinkey (the answer of which is a hink-pink itself) : Dastardly crop damaging beetle.
I’ll reword my other unanswered one, as I think “seller” is throwing people off.
Hink-pink: Where night sticks are purchased.