Cats can squirm out of some pretty cozy harnesses -- they are more flexible than dogs.
I don't know the brand name, but there is this excellent "jacket" type harness available, you'll have to check on the internet. I had one for my old kitty and it was comfy but no way she could get out.
The jacket fits like a little coat, those goofy things that vain pet owners force onto their pets, but this one has snug velcro belts that fit around the stomach and it's impossible for the cat to wiggle out. The leash snaps onto the top of the "coat".
Your cat may not like it but the cat will also be very safe and secure and can't escape to run out into traffic or some other peril.
However, your vet or groomer may require a carrier when you bring in your cat. I've used those "soft-side" suitcase style carriers with excellent success for my cats -- the soft fabric material seems to be less threatening than the hard shell plastic carrier. At least it seems that way. These fabric-style carriers also have a handy shoulder strap for easier carry, and your cat won't get bounced around inside the hard shell, instead a much softer interior with less bumping.
Most big pet supply stores should have both the soft-side carriers and the coat-velcro harness rigs.