A small switching transistor should do it. Look at a 2N2222 it can switch up to 800 ma. at 30 volts.
Since you have 12 volts to work with you can put the 2 LEDs in series for a total of 6.8 volts. You then need a resistor to drop the remaining 5.2 volts at .35 amps. Since 350 ma is probably the maximum spec. lets use 300 ma. so a resistor of 17 would be needed. Probably can't find a 17 ohm resistor, but 18 ohm is fairly common. The power would be I squared R or about 1.6 watts, so something like a 3 or 5 watt resistor.
So now we know the collector current (300 ma.) we can pick the base current that will saturate the transistor (30 ma) The base emitter drop is about .65 volts so the base resistor would be about 11.3/.030 or 376 ohms. use 390.