Either of the rival claimants could credibly take on the Monarchy ... which is why I would encourage one of them to rise up, claim the throne, and then publicly consecrate France to the Sacred Heart of Jesus. I think that enough would follow him to start shaking things up.
That's been the problem in countries like the US, lack of credible leadership, and when someone does rise up, they're either just taken down or else they're set up as controlled opposition to mislead the "rebels".
But in a former monarchy, any quasi-legitimate pretender could assume the mantle of power and get untold millions to follow him.