I'd get another 3/8 barb to put on the other end of the of the weldless cooler kit: that way you can use a 3/8 tube to go from the barb of the false bottom to the ball valve. Any tubing that's food grade and can withstand temps up to 212 will be good.
On Tap: Barleywine, Coffee Oatmeal Imperial Stout (big big beer)
Fermenting: Double Simcoe IPA
On the Bench: Racer 5 IPA
"One of the first things early caveman did, when he crawled out of the mudd, was to make beer. And cavemen everywhere are still making beer...and drinking beer." - Jean Sheperd