MarekB:
Video of SdFatTftBitmap example sketch demonstrating loading images from an SD card. The images were converted to RGB565 format using BMP24toILI565 tool (provided with the lib). One image loads and displays in around 190ms (using half speed for SD card SPI as the full speed did not work for me, probably because the SD card pins were connected to Due via a breadboard).
Most likely because the level shifting resistors on the TFT PCB weren't removed. you may want to give it a shot.
Now, full speed on a standard Arduino is 8MHz, that's not t even close to full speed on a Due, so I am curious what were the actual SPI clock multiplier settings.
Good stuff otherwise, with a bit of luck someone will port this to the ILI9327 3.5" 
BTW: Your first video doesn't work?