For anyone lacking a valid backup of their original Chrome OS firmware, I've constructed a ROM image with an invalid serial number from my C710-2847 (the launch model).
I've included instructions on how to modify the image to match your actual device before you flash it, which I strongly recommend. I can't predict how Google functionality may be effected by an invalid serial number or the default firmware hardware ID.
Also, please note that these instructions may be specific for the C710-2847 so pay attention if you have another model. There's only one recovery image for the C710, so I'm confident the firmware will work on all models. But, you may need to change more than 8 digits of the serial number with other models, IDK.
[Edit] If you have a C710-842G32ii, then I recommend using the BIOS.BIN contained in the shellball (which is also included). For an explanation & discussion of the implications - see the comments below - specifically, my conversation with Schokobecher.