1. Neither the US or British gold standards of the last 300 years, nor many of the others around the world, worked like this. Rothbard is just making stuff up.
2. A gold standard is NOT dependent on the amount of bullion in a vault. We saw that this was never the case. There were a few exceptions – China used Silver Bullion exclusively as money into the 20th century – but in the Western European world that was the rule.
3. "Defining the Dollar" at [Lewis' hypothetical value of] $1555 per ounce (from perhaps $350 per ounce when Rothbard was writing) is a devaluation. It's just the same as when Roosevelt "changed the definition of the Dollar" from $20.67 per ounce to $35 per ounce in 1933. You would think this would be what Rothbard and the other hard money types would want to prevent. (In fact the result of this devaluation was to make the US's gold holdings worth more than the monetary base, for a little while.)
4. This "100% backing" would be very brief. The normal operation of a gold standard would soon cause base money to diverge from whatever the bullion inventory happened to be. If you kept base money stable, then its value would diverge from the gold target. You only get to target one thing, value or volume, and the other is a residual. A gold standard is a value target, not a bullion reserves/volume target.
5. Although a small country, like Fiji, could implement some sort of "100% backing" system, there isn't enough gold in the world to do this on a global basis. That is why goverments tended toward "economizing on gold" for centuries. People who argue that it is possible that you could do it by "revaluing gold" fail to notice that this would be a devaluation. For example, let's say you "revalued gold" at $14,000 per ounce today. That would be a 10x devaluation of the Dollar. Eventually, prices would rise by about 10x. Then, you would need ten times as many Dollars to do your business. So, the amount of Dollars in circulation would have to rise, which would mean that you wouldn't have "100% backing" anymore.
6. What if the gold isn't there anymore?
Lewis prefers a Currency Board, like the one established by David Ricardo that revalued the pound in 1821 to its pre-Napoleonic-War parity. In Ricardo's own words, "it is only necessary that its quantity should be regulated according to the value of the metal which is declared to be the standard." 100% convertibility isn't needed. Interestingly, one of Lewis' further objections to the Rothbard plan is that it isn't stable enough! The initial target would have to be followed by a bout of floating as the called-forth supply of gold equilibrates with the new demand for it as a medium of exchange. Lewis pegs that as a devaluation if gold risies higher.
The trouble with Lewis' plan is that he's arguing in a circle. How is the value of gold determined? By the market price. How would the convertibility ratio be determined? By reference to gold's market price. What if that market price changes once the Currency Board sets its own price? What then? How would a gold drainage be prevented, except by setting the price higher than the ratio warranted? Doesn't that mean a last one-shot burst of inflation? And what's to stop Gresham's Law from asserting itself if the inflation carries over after the Board begins operating? Since Gresham's Law applies to currencies whose prices are fixed, what will happen once it asserts itself? Will the Board have to outlaw the private possession of specie? If so, then what kind of gold standard can it be?
There's only one way to break the circle: redefine the currency to be a fixed weight of gold and nothing other. That approach, though, leaves the rather large issue of what will happen to the extant greenbacks.