A single barrel is more accurate and I don't believe that multiple barrels are required for a gatling system to work in this application.
Recoil and heat of the high RPM are less of an issue as the requirement is not for a sustained automatic rate of fire. Recoil does have an effect on accuracy however.
I imagine a gatling system where only the .556 chamber revolves, like in a revolver. The chamber/cylinder will hold the # of rounds in the burst (3?), and the entire mechanism will recoil back in the housing on rails like an artillery piece, for the length of the burst, where, at the end of its cycle, the entire chamber is reloaded from the feed device and the barrel & chamber mechanism are returned forward into battery.