It's very unlikely that if France intervenes, it does so alone. French public opinion and political circles were divided over the question, up to the point the Quai d'Orsay (foreign ministry) did it best to prevent this from happening, and this is a period where the Emperor seeks to build political consensus by liberalizing the regime. In this context, an intervention should be decided only to support the British in the event they engaged. Look at the example of Mexico, it begun as a multinational intervention, not unlike Crimea and China, before Spain and the United Kingdom withdrew after reaching some understanding with Juarez.
Politically, the only military action that Napoléon III could justify would be to protect maritime trade and break the blockade, while indirectly supporting a Confederate reconquest of New Orleans and the Mississippi delta.
