Here is an adorable baby. She's about 8 months old in this picture, and she rides in a Graco Snugride. Here she is wearing her coat. The coat is actually a fairly thin fleece, but it's a size or 2 too big (I think it's actually cut large).
Here she is, wearing the coat in her carseat. I've buckled her in, and tightened the harness as much as the coat allows me to.
Here, I took her out of the carseat, without loosening the harness. I took her coat off, and put her back in without adjusting the harness.
When a harness is properly tightened, you shouldn't be able to pinch any of it. Your fingers should just slide off. You can see in this photo how much I am able to pinch
A thin fleece that is too big is just as bad as a puffy coat. A thin fleece is safe, as long as it fits the child snugly.
Here is a good option. There are carseat covers that go over top of the carseat like a hairnet. These don't interfere with the harness, and don't prevent you from getting the harness tight enough.