Hi!
I found a fix to our issue! Instead of attaching your SD Card Shield to your board, go ahead and just lay it next to your Mega. Next, using some standard jumpers, connect pin 11 (SD Shield) to pin 51 (Mega), then connect pin 12 (SD) to pin 50 (Mega), then pin 13 (SD) to pin 52 (Mega), and finally; pin 4 OR 10 (SD) to pin 53 (Mega). After those four connections have been made, connect 5V from Mega, to 5V on Shield, and connect GND from Mega, to GND on shield.
EDIT: Didn't see your reply, you don't need to bend anything as long as you follow what I did above. You won't be able to use the pins on your SD Shield, but who cares, they're not taking up any space on your Mega except the 6 I told you to use. If you want though, you could bend the pins 4 or 10 and 11,12,13 pins out of the way, and wire it to the arduino, but if you ever plan to put the SD shield back on the Uno, you'll have a bad time. Also, I changed both to 53 and it worked.
pinMode(53, OUTPUT);
SD.begin(53)