What happens if two chips drive the same line is "who knows, but generally nothing good". Essentially you get a short circuit from Vcc to ground through the high side transistor of one chip and the low side of the other. One of them might win, or you could get an intermediate voltage. Power draw goes way up. You can also get weird time dependent glitches. It's possible to damage one or both chips, although malfunction is more common than damage.
Some SPI flash chips have a RESET pin with an internal pullup and might be left floating on the PCB. If you are lucky enough to have that you can just force the target chip into reset while your mod chip is installed.
Its possible if the target chip has relatively weak drivers and the mod chip has beefy output stages that you can just overpower it and things will work. It's not really ideal, but it can work. Disabling nCS would be work very well but is generally just as hard as lifting MISO.