I have no problem with you all divvying up the ice moon. But I've had a "Standing and official land claim on the lava moon." since moon balloons. No different than my owning any other piece of land, only the moon is in space and not easily habitable.
America has some claim to the moon I'd say. They just haven't got the money or the want to do anything with it. As far as Antarctica, there's oil under there people want, which is why it's been doled out, and if say, Holland, put their flag there a couple hundred years ago, nobody took it from them, and they never abandoned it, the Dutch would own Antarctica the same way America owns any one of its states. Avalonia's ownership of the lava moon is no different.