You do not need to license the trailer if it is registered in KY (or any other state that does not require trailer license). Just be sure to have proof of the KY registration in case you are checked. I verified this with the IN State Police. They could not realistically enforce a license plate law on every out of state vehicle that passes thru IN.

INDNR requires the $20 lake permit on ANY boat on state-owned waters (such as Patoka). They can be purchased online or at any DNR office. The Patoka office is located at the entrance to the state park. Sometimes, they will have someone at the ramp selling them as well.