No, you don't have to have a permit. Follow this link.
http://www.api4animals.org/b4a2_exot...state.php?s=ky
Here's what it roughly says.
301 KY. ADMIN. REGS. 2:082 - Transportation and holding of exotic wildlife. (4) A person may temporarily transport or display a prohibited animal listed in this section through the state for less than ninety-six (96) hours if at all times the animal is maintained within a confinement sufficient to prevent the animal from escaping.