As planned I took apart the saucer chair. I added a second layer of batting and cut off a half inch all around. I sewed it back together, replaced the bias tape (sadly not quite as beautifully the second time around) and replace my zig zagged tuffting. The tuffting was a great improvement over the first try.
Here's the new chair, though it's hard to tell in the picture it really is a big improvement. It now looks soft and comfortable and less floppy around the edges.
Here's a photo of the original, before the addition of the second layer of batting. See how this one is all smooth but the above one is all puffy?
And here's a new picture of Catie in the chair, just because she's adorable.