Hmmm....interesting. While I haven't yet thought of HOW such a system would come about, I think there might be two methods that the French could use.
#1: Seeing that I'm a hardcore fanatic of the Holy Roman Empire, possibly second only to the legendary Susano himself, I suppose you could have something similar to the Prince-Electors, only instead of having seven as in the HRE, you'd have three here. Each royal house would be free to use any method it wanted for selecting its heir/elector, and the three electors would choose the next king from their number. Since there's only three of them, there might have to be a rule that no elector could vote for himself, otherwise you'd probably just end up with a 1-1-1 tie most of the time... So whichever elector can garner two of the three votes would become the next king.
#2. Maybe a "rotating kingship", just like the President of the EU Council, except in this case it would obviously be for life instead of a fixed term. Just as in #1, each royal house would be free to use any method it wanted to pick its heir. Since the Bourbon line is arguably the most prominent, perhaps the current Bourbon heir would become the first king. Then whenever that Bourbon king died or abdicated, whoever was the current Orleans heir at the time would become the next king, and when the Orleans king died, the current Bonaparte heir would become the next king, and upon his death it would revert back to whoever is the current Bourbon heir, and the cycle starts all over again.
I actually like this idea a lot...a compromise between the royal houses like this might be just what the monarchy needs to put an end to all those pesky pretenders to the throne that threatened its stability. Thoughts?