It's a mix of 25x shenanigans and upper playfield work. The Lannister castle effect gives advances on the Lannister value similar to hitting gold targets, but it can be built up to the cap of 75M. Since Lannister saves mode progression, any advancement of your valuesare also saved. Then it's hitting the ram to light 5x Playfield, restarting Lannister, and then hitting shots in combo to clear the mode.
Safest way to do it is:
1) Play Lannister while not hitting any lit shots. Use the upper playfield to build the mode scoring to 75M a shot. Use any extra time to light shots (optional).
2) After Lannister ends, hit the battering ram to start 5x Playfield. If for some reason you accidentally hit the mode start, sacrifice a danger to keep the "Choose your Battle" lit while skipping.
3) Start Lannister, and collect shots with the highest combo values you can (Tyrell helps a lot). 75M X 5 playfield X 5 combo = 1.875B per shot, for a total of 9.375B on the mode max. Without Swords to boost combo X (as an opening move), you can max out at 5.625B.