Hi Franco,
Yes it is possible with LPC1768 although throughput could be limited by SPI interface (or does it support 4 bit I/F?).
I went through a similar exercise with a NXP Kinetis MCU (K22). I got a FatFs demo running that connected to SD card via SPI. Then I got a demo working that mounted a small RAMDISK as a USB Mass Storage Device (MSD) - aka Mass Storage Class (MSC). Merged these two demos and then figured out the glue code so MSC callbacks accessed the FatFs diskio API (instead of RAMDISK buffer). This also needed state logic to ensure one owner the FAT file system (host PC or local FatFs MCU). My app only starts the USB device when user select switch is in OFF/CONFIG position. The USB device is only stopped after MSC activity (E.g. events/callbacks) cease for 3s (e.g. user ejects the MSD). Only use FatFs to access the SD card when the USB device is stopped to ensure integrity of the FAT.
I used NXP SDK v2.x for this, but I don't think the LPCxxxx range is supported yet so you might have to look for other middleware (LPCOpen?). Or, as Zack said, MDK professional should also have all the middleware and drivers for this.
Since throughput is important to you, check if you MCU supports the 4 bit SD card interface (e.g. SDIO, SDHC, MCI, etc.). Trick is to find middleware with file system support, drivers and and working demos. I have seen demos that mount SD card as MSD with 4-bit I/F but not for the MCU I was committed to. Even with such a demo you still need to merge USB MSC and to ensure single ownership of the FAT file system.
Hope this helps,
Sebastian