I think the reasoning here is too specific. Most currencies are only inflationary long term. If during a 10 day window a trillion euros flood into the US dollar market, I doubt the Fed will be able to respond fast enough to keep the prices fixed.
Second, even if they didn't the Fed could easily respond in real time. It's not as if the Fed actually, literally "prints money". They would just go onto those forex markets and sell people dollars, transferred digitally and made up out of thin air. And their servers wouldn't crash while doing it.
Edit: I should add to this that, at least to a first approximation, the Fed is not terribly concerned with the nominal exchange rate. They are charged with maintaining full employment and price stability within our own economy, while the nominal exchange rate is allowed to float against other currencies. Because have have our own production economy, there are good fundamental reasons for fluctuations in the real exchange rate that should be permitted to occur.