I would definitely use another regulator to supply the LEDs. THe big issue is the onboard Arduino regulator is Linear, which means the more current it draws, the more power it burns off as heat. This means it can get really hot with large currents, and the 1A rating doesn't quite say how much heat-sinking it needs to work at that, and the Arduino's regulator's heat-sinking is limited, since its just mounted to some copper on the board. A better idea would be to have either an external switching supply to supply power, multiple LDOs, or a switching regulator. Pololu sells switching regulators for $5-7 Pololu - Regulators and Power Supplies. Just make sure you get one with enough current.
You can still run some stuff off the Arduino's regulator, but you should at least run the display LEDs from an external regulator/supply.