Doubts about the supply. 12.5~14.5volt seems to be a battery charger, unreglated.
Sounds to me just like the voltage in a car
A normal NPN indeed needs a to high base current. A logic mosfet is indeed a easy option.
This TIP120 works as well. A base resistor of 1k to 1k8 is fine. You have a Vce of around 0,75V so a power dissipation of around 300mW. But 0,75V is very significat to the 14.5 - 3 x 3,4V = 4.3V left for the resistor so pich the resistor accordingly.
But, like I said, the source sounds like a car. The brightness difference between 12V and 14,5v will be big. For 12V you have (12V - 3 x 3,4V) / 220ohm = 8mA. (Okay, voltage drop across the led is probably a bit lower so around 10mA). And at 14,5V you have (14,5V - 3 x 3,4V) / 220ohm = 20mA so roughly double the current! And when you consider the TIP120, lets calculate the resistor for ±20mA: (14,5V - 3 x 3,4V - 0,75V) / 20mA = 178ohm => 180ohm. At 12V we now have (12V - 3 x 3,4 - 0,75V) / 180ohm = 6mA, that’s only a third!
So if you want to use it in a car you might want to consider to change it a bit. An option would be to only put 2 in series but then you wast a whole lot more power! You can also regulate the power to the leds to lets say 12V, no matter the car voltage with a DC-DC step down converter. (Or use 3 or 4 of them to keep the load on each low). Then you can pick a resistor for 12V and the light will stay the same, no matter of the motor runs or not.