He didn't serve the Bourbons because he didn't respect them and because of the way they treated him and the returning garrisons from outside the country, especially the victorious ones, such as his, Lemarois' from Magdeburg, and Carnot's from Antwerp. Davout was not even 'permitted' to ride with his corps on the way home from Hamburg, the accusations from the Bourbons, in part, read that he had 'committed arbitrary acts which tended to render the French name odious.' You expect Davout to respect and serve such 'waiting gentlewomen' (in the words of Hotspur)? Davout, and many other like him, had nothing but contempt for the Bourbons.
The Bourbons had no idea had to treat veteran soldiers, and completely had no idea of Turenne's old adage about how to lead them.