I would concentrate on keeping the combo multipliers running (the red flashing arrows on all the major shots) if those time out, you start back at 1x. So, if you are going to make the final shot of Stark and brick a couple times, I am sure those will have timed out, and if you don't have any playfield multipliers going, your value will be next to nothing.
I have found that what is the best way for me, is not the best way for others. I have been exclusively been playing as Greyjoy recently (at home and in tournaments).
I start the game by hitting the stark ramp then lock targets. Lock ball one. hit stark ramp, then lock targets. Lock ball 2. Do that a third time to start MB. Start Lannister mode. after MB is over, play stark. If Martell or Tyrell (green) isnt lit yet, work on those. Typically if tyrell is not lit yet after your first MB, you end up getting to the second BW MB by trying to light it.
So yeah, for greyjoy I take the same houses into HOTK. Although, having to do those stark bonus rounds kills me a lot and makes it really hard to last long enough to make it to the big hurry up pay off.
Another way to do it is to punt on Stark and just take Martell, Tyrell, Lannister, and Greyjoy. that is a quick route to the big Super Jackpot hurry up. Yeah, you have to hit each shot twice, but they are easy shots and you also have the add a ball to use. then save the stark mode for after HOTK.