Google AI gives an interesting answer:

The person you're likely thinking of is Albert L. Brown, a Prohibition Agent shot and killed in 1930 in Kentucky while trying to arrest bootlegger Rodney Selby for selling liquor (or "likka") to agents, with Selby getting the fatal shot after selling them "four shots of liquor". "Paducah Brown" isn't the name, but Paducah is a major city in Kentucky where the ATF has a strong presence, and Albert Brown died in the line of duty in that state, fitting the description of being shot over moonshine/liquor.

Adding the word "song," I got this:

The song you're likely thinking of is "Paducah" by The Handsome Family, which tells a dark, folk-story about a man named Paducah Brown who gets shot after selling "likker" (liquor) in a tough town, capturing that classic eerie, narrative country/Americana feel. It's a well-known song in that genre, fitting your description perfectly.

I looked through stuff on "The Handsome Family," and I couldn't find anything that fits the description. Please post what you have so far, and maybe we can help ou find the rest.

Joe Offer, Mudcat Music Editor