The Lizards faction bonus got me thinking about a possible new card the Parliament could generate:
Strategy Card - Diplomacy: Goodwill Gesture - Increase the Support Cap for one battle by 25/50/75/100.
Although that does step slightly on the Lizard Factions toes. There could theoretically be a sabotage card that decreases that for the enemy, but just the concept of that sounds too overpowered and unfair.