I think he'd likely use Jean as his name as Kellan said, as there was some history to it and there had been previous King Jeans.
The French Princes could be very picky and didn't necessarily like the idea of their sovereigns having 'odd' or untraditional names. The row that broke out when Anne of Brittany had her first son comes to mind, wanting to name the Dauphin Orlando. Obviously as you move into the 17th and 18th century, you do see more unique names pop into the French Royal Family, but they're typically prefaced with the more traditional names (Louis, Charles, Philippe, Henri). Jean had fallen off as a name for French kings, but it at least has a history: Gaston and Baptiste do not, and Gaston especially as having rather Italianate influences would definitely be a no go.
He wouldn't be the first king to take a different name when he became king.