This came very close to fruition in 1862, and I could see a wily UK playing the US and France off of each other to whittle down competitors.
France has a larger navy but no bases for transAtlantic operations, they also will have interests in Mexico that they may try to defend. Union land army is larger and this might be the impetus they need to acquire Henry rifles as stanard firearms. In this case, they would get a serious edge on the battlefield and would try to drive out the CSA and the French at the same time. The US navy is also growing quickly and has ironclads at her disposal, which were *not* transoceanic vessels. France can send wooden vessels out but can they beat an ironclad fleet?
CSA gets supplies and extra manpower, perhaps enticing one or two more border states to secede, but I still think that eventually the rebels fail and now the greatly enlargened US has a bone to pick with France. Bismarck sees this and Germany becomes much better friends with us, as we take Franch colonies and they march on Paris.