There's nothing computationally expensive about it. The Total War series of games run thousands of discrete animations at once. This is just another animation: a bog-standard application of matrix mathematics, which is equally doable on CPU or GPU.
Probably the only reason you don't often see it is simply because most people's breathing is not that noticeable at the distances you tend to see characters from in games, i.e. several metres. In other words, it hasn't been added to many games quite simply because it doesn't add a whole lot. Chest displacement of a centimetre or two is not that noticeable, and slow breathing (which happens most of the time) is even less noticeable.
Movies are high-fidelity, especially the stuff seen in Pixar movies, so it's quite probable that they'd give us much attention to fine detail as possible. Production processes in games tend to have other goals, and keep visual detail as reasonably simple as possible to reduce the cost of rework wherever necessary, while still looking pretty good. Merging different animations / transforms can cause issues, so it's better to keep things simple, by and large. (But I doubt this applies much, here.)
Conclusion
Quit worrying about small factors and just write the code. If most of us were to sit around worrying about tiny issues like this, we'd never get anything done. It's a merged animation, and not a particularly complex one as these things go. It's not a big deal and has been done many, many times before. If it fits your style, great, do it. Just because others haven't included it in their games, doesn't mean there is any overarching technical limitation involved.
