This particular library won't work for you. They are using the SPI bus of your arduino to shift out the data.
If your other hardware doesn't use SPI move it to other pins, and free them up for this setup.
If your other hardware does use SPI then all you really need to do is connect the 595 to the same SPI lines, and choose another Latch pin (say D9).
This is because the SPI bus can support more than 1 device.
You can also try the FM NewLiquidCrystal library, I'm pretty sure they have support for a 3-wire SR which can be driven by any pins you want.
Here's some more info on how/why you would want to use SPI:
SD cards use SPI as well. So basically you can use the same SCK and MOSI pins for the 595 (just before the resistors), and a separate chip select pin. For example D9 or whatever you want.
Wow very nice article indeed. I will try as soon as possible.
Also PCB images are very nice. Did you use 2 PCB's for this project.
I used a single 24x18 perfboard to build the prototype. I'm now waiting for professionally made PCBs to get here. I already made a new PCB design which is much more compact, so should be considerably cheaper to produce. (If there will be any user demand for these)