Best I could do tonight was 3:05. I'm actually not finishing any modes with the upper playfield. I think if I could get better at the upper playfield I could beat my 2:48. So right now my strat is almost exactly what chuckwurt said.
Begin as Stark. Play Stark with minimum shots (3 center, one orbit).
Loop center and begin Lannister, kill with button. Immediately drain.
Ball 2 get Martell lit with 3 right orbits. You don't want to wait till 4th mode to do Martell because EB will be lit and that kills your time since you can't speed past the animation. Start Martell, kill with button, immediately drain.
Ball 3 get Greyjoy lit. Hopefully you don't need all 3 left orbits by this time. Start Greyjoy, kill with button. Shoot center to begin HOTK.
Sounds easy right?!?
As soon as you accidentally hit a green target you get slowed down by a lock sequence. Hit enough of those by accident and you start Blackwater which just kills your time.
Another strat is to begin as Targaryen. Now you only need to finish 3 more houses. Do Stark, Martell and Baratheon, since those can be finished quickly. I had some 3 to 3.5 minute runs doing this. If I could get better at finishing with the upper playfield this might be the better strat. You don't have to drain and that saves time since technically you could do the whole thing on one ball.