No charge for the bike case as such, generally. What they will undoubtedly try to nail you for, though, is total weight of luggage. I've travelled lots within Europe with bike/hard case, and have run into this repeatedly.

Split your total weight carefully between your case and your main suitcase, so they're roughly balanced (lower or more strictly enforced per bag limits there). You can often get cut some slack if you show your inbound/homebound air ticket from the States, and plead that this was treated as acceptable total weight for intl. travel during your trip over (so why OK for US to France, but not France to Spain?). Good luck. Like so much, it's really up to the agent/airline.

One other warning: carry-ons are routinely slammed for overage within Europe, and the size/weight limits are far, far smaller than in the USA. Count on a 5 kilo (11 lb.) max for your carry on. Unless travelling in business class, have often been demanded to check my briefcase. Ouch.

Another: forget about taking it on most French trains, including most intercity TGVs. Check ahead of time.

Again, good luck.

PS -- a story I can't resist: once watched from the departure area forlornly as my custom Spectrum Ti was quickly loaded onto the wrong plane into Lugano, the hatch was slammed, and the small Saab jet taxied off for takeoff to God-knows-where. They had only two planes in the tiny airstrip (and probably a total of six bags), and still got it wrong.

It being Switzerland (albeit, Ticino), when I hollered, they immediately radiod the plane from the tower and aborted the takeoff in time. The airport's Swiss-German boss just smiled apologetically, shrugged, and said "hey, silly Italian Swiss -- what are you going to do?!?!"