A blendshape will not work, because it is not actually transform data. It is a deformation just like the smooth skin you are trying to influence. Influence objects affect the skin based on actual tranformations.

Rubberkidney's suggestion is a good one. It's a brute force method, but it works (I've used the same approach on my rigs).

More often than not, the reason the breathing doesn't look right it probably because of how you are animating it. Breathing motions are not restricted to the chest area. It combines subtle movements from the entire upper body. In a normal human, the shoulders move up a tiny bit, and there is some movement in the upper spine.

For your character, you might try some overall body movements to accentuate the breathing. Maybe have the body shift slightly up and down. Do you have a render that we can look at to see what you have so far?