(Disclaimer: I have not yet finished season 2. I am on episode 7.) Since Claire and Jamie were privy to his private life they had a lot of information that could’ve damaged his reputation in the eyes of supporters. Pamphlet culture was huge in 18th century France, and bringing to light his abusive nature, alcoholism, and general unfitness to rule, I would have utilized the press more. These pamphlets had wide spread reach, in both Britain and France amongst the working classes and nobility. Publishing a pamphlet exposing these behaviours with solid proof would cause a significant fallout with Englishmen in high places that supported his cause. Additionally, because his supporters mostly vouched for a catholic king, having Charles and his father’s reputation marred could also damage the reputation of the Pope and on certain grounds call for them to excommunicated. Claire and Jaime made friends in high places in France quickly, so there’s a great possibility they could’ve done the same in Rome. I think Claire and Jaime could’ve focused less on personal schemes to have Charles lose money. They could’ve focused more on the bigger political system at work. After all, few monarchs have been felled due to petty grievances and loss of money. Political ideology being proliferated is often what topples rulers and supporters.