We got word back from our RF expert, and here's the simple version, which isn't much different than we thought:
A perfect circle ground plane is great at one frequency since all edges are the same distance from the antenna. It's awful at all other frequencies. Unfortunately, you need to get this just right, and with a transponder, we transmit at one frequency (1090 MHz) and receive on another (1030 MHz), so there is no perfect circle that meets both frequencies.
We actually like the octagon idea, because it varies the distance between the antenna and the edges, which gives it good performance over a range of frequencies, but so does the rectangle/square that Dynon recommends. We REALLY don't like the idea of a circle, and disagree with the Kitplanes article that an octagon and a circle would likely be equal performers. They are if you cut your foil by hand with a pair of scissors and don't get a lot of accuracy. They aren't if you go break out the laser cutter and make a circle with 1mil accuracy, like some people would inevitably do under the false assumption that accuracy was a good thing here.
Next comes the idea of the size. You really do want a ground plane that is either huge, or specifically sized to the frequency you are using. If 120mm is the right size, 240mm is really, really the wrong size. One of the issues here is that the structure of the plane can interfere and/or change the proper size. So unless you can test and refine the install, there's no easy way to say what the right size is, but "huge" is always a good bet. So we recommend 120mm (which actually varies a bunch because it's a rectangle) or greater than 700mm (which is just "huge").
Thus our recommendation that you not make a circle, and you make it either a specific size or big. It provides most of the theoretical efficiency of a perfect ground plane without the caveats of needing to get a perfect ground plane, well perfect. One of the worries is that you actually want a rough, "inaccurate" ground plane, which can be hard to convince a homebuilder of

We really like Mr. Weir's hand cut, visually non-symmetrical, foil ground plane. That's what we'd call perfect for the application at hand.
Now of to the kitchen to get some aluminum foil, scissors, cardboard and double sided tape!