One thing that might help you is to check out tutorials or patterns for top down hats. It's the same principle, knitting in the round. With those hats you keep increasing, which you'd have to do with the coasters, and it comes out lying flat. When you stop increasing and knit straight, the sides come in and it's the straight down sides of the hat. If following hat directions, you would stop when it gets to the size you want. I would suggest pulling the tail on the first round or two to close the hole.
Have you tried crochet for the coasters? In making a hat in crochet, you crochet in a circle. You keep increasing. It looks like a flat plate. You stop increasing at a certain point to make the hat curve inward and downward. I think crochet would be better for something like this.
Welsh corgis are cool.