Not well, but House Lannister would be more difficult to deal with than the Rogare Bank in Lys – it’s a powerful military sovereign, its vassals are quite cowed and loyal, it’s rather far away, and there aren’t convenient rivals to support.
My guess would be that they’d have to resort to more indirect methods – providing support to the Riverlands, Reach, etc. to build up anti-Lannister forces and contain the Westerlands, pay the Ironborn bonuses to raid Lannisport commerce, and wait for an opportunity like the War of Five Kings to get the House declared traitors.