Is it possible to overwrite the system factory reset option with my own backup?
When you start the factory reset in the settings it just runs recovery which the actual resetting is done by. Normally, recovery just wipes everything in /data and /cache in all (and /sdcard in most of) phones, since that is all a non-root user can modify.

So a factory reset is not a fresh install of the ROM. It is just the wiping process of every made-by-user change on the phone.

I want to stress it since as much as I understand you think factory reset does some sort of overwriting. It doesn't do that. Vice versa it wipes stuff. It, by default, doesn't restore, install, overwrite anything.