Hi Vitor,
No, it is not using the Kinetis SDK, as the version is set to 'none':

That component is only there to make the bridge for either 'none', SDK v1.3 or SDK v2.0 and to provide the necessary wrappers.
As you should see with the example, you do not need to use the Synchromaster methods directly: simply use the FatFS functions.
In that example I'm using FreeRTOS because it makes the application scalable. But if you want to run it baremetal, here is another example (well, for a different board, but you could use the same thing for the FRDM-K22F and KDS):
https://github.com/ErichStyger/mcuoneclipse/blob/master/Examples/CodeWarrior/TRW-K60N512/TWR-K60N512...
Erich