From my experience, after traveling with a lot of different sizes of Camera gear:
The airlines do not want you to check in your camera gear. If something gets damaged, it always gets messy and expensive. Not only do they face arguments about the gear itself, but also often about the whole value of the trip since the photographer basically goes to a location for nothing if the camera gear does not arrive in one piece too.
So as long as the additional luggage contains in the majority camera-related items, they are fine. What they do not like is if you come with a huge bag which is filled with all kinds of stuff and then a compact point-and-shoot to justify the term "camera-bag". Similar things happen with Golf and Diving gear in many airlines (depending on the destination though). They travel with reduced costs or even free beyond the normal travel allowance.
So as long as you can argue that this whole bag is used for camera equipment, you should be fine until a certain dimension, but, as mentioned above, it highly depends on the mood of the check-in personnel since there is no clear rule on how much you can actually carry. Good Luck!