Closest public launch that I can think of would be at the freelaunch there on the highway past Sharps Chapel. I guess you could put in at Palmers or maybe even at Lost Creek. IDK if they would let you though.
To be honest Im not sure if Lost Creek has a ramp or not. Its right there on the lake and people has boats up in there everywhere. Best bet would be the Claiborne free launch. Coming out of Sharps Chapel take a left at Hwy 33 and its about 1/4 mile if that on the right. Cant miss it.