Well, from my experience with the C280, it seems to be a warmer running car than a lot of others.
A long climb like that puts a lot of stress on the engine and can start to overload the cooling system. The engine is fully insulated on all four sides and there is not a tremendous amount of space under the hood, which makes it harder to pull air through the radiator and dissipate heat, the radiator is also not exactly the largest thing in the world.
As far as the transmission, it could overheat, I would check the fluid, and if it is burnt, replace it soon. I always thought a transmission temp gauge would be nice, unless there is some way to get a tranny temp reading from the climate control. I know that on the 722.6 tranny, the transmission temp needs to be monitored through an internal temp probe in the tranny to get an exact reading. Does anyone know if this is this a value that can be pulled out of the climate control.