The Mega's voltage regulator is very likely working but overheating due to the current drain and going into thermal limiting mode. In particular, the back light on the 12864 needs quite a lot of current. Also, some 12864 displays have the back light series resistor built in, while others need an external series resistor.
I recommend driving the back light through a suitably large series resistor from the 12V supply, so that it doesn't add to the power dissipation in the 5V regulator. From memory, I think the backlight of a 12864 is rated at 100mA and it drops a little less than 4V. So a 100 ohm 1W series resistor should be about right. Or about 200 ohm 0.5W if you are OK with lower brightness.
I recommend driving the back light through a suitably large series resistor from the 12V supply, so that it doesn't add to the power dissipation in the 5V regulator. From memory, I think the backlight of a 12864 is rated at 100mA and it drops a little less than 4V. So a 100 ohm 1W series resistor should be about right. Or about 200 ohm 0.5W if you are OK with lower brightness.