Rather than putting a cat in a carrying case, simply snap a handle on them, pick them up, and go.
A cat handle is a tube that goes around the cat torso, supporting the entire length of the cat body. Their head and legs are exposed, though there is a support band around their chest. Think of it like a hard-cased pet-sweater, without sleeves.
The case is made of hard plastic, so as not to pull the cat up uncomfortably in the middle when lifting it by the handle.
Easy to use: the tubular case opens on a hinge so you can place the cat inside, then snaps shut on top. There is one large plastic handle centered in the middle of the cat's back.
Pick up and go!