Since the State of Kentucky requires you designate a party when you register to vote, ....
That statement got my attention, so I pulled up the KY Board of Election website. Here's a FAQ that clarifies this issue (bolding mine):
Q: Must I choose a party when I register?
A: Kentucky has closed party primary elections. You must register as a Democrat or Republican to vote in that party's May primary election. If you choose a different party registration other than Democrat or Republican, you cannot vote in their respective primaries, but may vote in nonpartisan city and judicial primaries.
All eligible voters can vote in the November general election or special election.
Whew! The mandatory party registration only applies to primaries.