The FDIC is unlikely to fail. After all, as long as there is no major credit crunch (de-leveraging), money simply moves from one bank to another, and insurance premiums are collected regardless of where the money is located, money (and MTM short-falls) can be absorbed to big banks. Big banks, Fed and FDIC can work together to sort things out like this time.

Additionally, the dollar is merely a piece of paper, and the FDIC essentially guarantees pieces paper with $250,000 or whatever face value printed. There may be intricate mechanisms at play, such as the FDIC issuing bonds which the Federal Reserve subsequently purchases, or the Fed buying Treasury securities and providing the proceeds to the FDIC in the form of loans or capital injections. Or Fed directly offers credit line with failing banks. It is difficult to envision any issues the FDIC could encounter that could not be effectively addressed by the Federal Reserve essentially printing more money.

The value of US dollar is all about confidence, in the event of failing of FDIC, it is a testing moment of the confidence of USD. Although forced to print a lot more money could be bad for confidence, but not that catastrophic as FDIC failing.