If the C key height is indeed the source of the problem (which it very well could be), you will need to either shave some cork off the key's foot (which regulates the height, no bumpers on this model), or if that isn't enough, you will have to remove the key, and VERY CAREFULLY file or grind down the foot a bit and replace with a cork or ultra suede silencer of suitable thickness. The way it is designed, I wouldn't try bending either the key foot or arm. A tech might unsolder and move the angle of the foot slightly on the hinge rod if this is grossly out of alignment/adjustment.
My Zephyr alto's D is right in tune and speaking well with the C key about 10mm from the edge of the tone hole to the bottom edge of the key cup rim at the middle of the cup (I'm assuming that you don't have a key height measuring gage).