SdFat does not check the clock speed. The only assumption is that millis() works correctly.
SPI_FULL_SPEED sets the SPI clock divider to 1/2 so the SPI clock speed in your case is F_CPU/2 or 10 MHz.
You can't hack software to get the equivalent of a 16 MHz board since the the hardware SPI clock divisors are powers of two.
SPI_FULL_SPEED is F_CPU/2
SPI_HALF_SPEED is F_CPU/4
How does it fail?
If it fails in the init() call try changing SPI_SD_INIT_RATE to 6 in SdFatConfig.h
//------------------------------------------------------------------------------
/**
* SPI init rate for SD initialization commands. Must be 5 (F_CPU/64)
* or 6 (F_CPU/128).
*/
#define SPI_SD_INIT_RATE 5