I had never heard of this method of removing carbon so the question was interesting.
After doing a bit of looking I came to some conclusions.
First, the decarbonizing that is most often mentioned is for two stroke engines, not four stroke engines like the RE. As we all know, two strokes can build up some really heavy layers of oil soaked carbon.
Second, a claim that sodium hydroxide (lye) would dissolve the carbon seems to be a myth because although it reacts with carbon dioxide, it does not do much of anything with pure carbon. It will rapidly eat your skin, hair and eyeballs though, but that isn't due to the carbon in those things.
I suspect the reason the guys who have used lye to clean up their two stroke exhaust systems saw the carbon coming off of the pipes is due to the fact that lye totally dissolves oil and the oil in the oily deposits from a two stroke engine would be removed.
Once these oily deposits are gone the carbon that was imbedded in them would easily be washed or scraped away giving the illusion that the lye dissolved or somehow weakened the carbon.
Getting back to a Royal Enfield the only time this method of removing carbon might work would be if the bike was burning a lot of oil and if it was doing that, carbon in the exhaust should be one of the last things to worry about.
I don't think messing with caustic chemicals and potatoes would improve the exhaust flow on a Royal Enfield and any small benefit is certainly not worth the risks of messing with them.
OK all you lye water and potato lovers, have at me.