I hadn't considered level shifting, that will probably be necessary as the D0/D1 pins are 5 volt logic. It's ironic that the Linux side is 3.3V logic, and there are already level shifters in place to translate from the D0/D1 pins to the Linux processor. It would be nice if shifting from 3.3 to 5 back to 3.3 wasn't necessary, but that would require trying to solder fly wires onto tiny pins, which can tend to be finicky and fragile. If that were done, it would probably be necessary to disable the existing level shifter on the board, which I believe can be done by controlling a GPIO line on the Linux side (the proper pin can be determined by looking at the schematic, which I don't have in front of me at the moment.)
The SPI interface shouldn't be necessary for proper functioning of the hardware or the Bridge library. As far as I know, it's only used to load a new sketch into the '32U4 processor using the Linux system (either from the web interface, network socket, or Linux command line.) For example, if you choose the Yun's network address from the Arduino IDE's Port menu, when you upload a new sketch the firmware image gets sent to the Linux processor, which then loads the image into the '32U4 using the SPI interface, just as if you were using an ISP programmer pod.
If you wanted to use the SPI interface in your project to communicate between Linux and the Due, I'm sure you could hook them up, but I wouldn't count on being able to load software over it: the command line tools on the Linux side to do so assumes a '32U4 processor, and would take some changes to be able to load code on the Due.