The more I thought about it, the more the dart idea seemed like a bad one. I got out my seam ripper (sigh) and unstitched all (double sigh) the log cabin blocks down to their centers, while watching screechy singers on American Idol get standing ovations. I restitched the blocks, trimmed them so they were all the same size (no easing needed, see?), sewed them together, and voila! The quilt was perfectly flat, no easing, no hills, easy to quilt, no tucks or gathers. No puppies needed.
But when I stood back and looked at the finished quilt. It seemed a little somber for kids. The puppy appliques now seemed like a great idea. I fused on five fuzzy puppies (fusing is not easy with fleece, but I did it primarily for basting, so a good solid fuse was not needed), added free-motion reinforcement on the edges, and now I'm happy. On to the next non-flat fix-a-quilt project.