If a meter is the same thing in all possible worlds, we're referring to an abstract measurement, divorced from any definition that can vary.
The meter has been defined as a tenth of a millionth of the distance from North Pole to equator going through Paris, or by a platinum-iridium bar with scratches on it, or as a certain number of wavelengths of a certain krypton emission line. It's currently defined as the distance light travels in vacuum in a certain fraction of a second.
We're not given an abstract standard Earth that will be the same in all possible worlds; it's easy to envision a possible world where the planet is somewhat different. The distance between two scratches on a metal bar will vary with temperature. It's at least conceivable that the krypton emission line frequency could be different, along with the speed of light, in possible worlds. (The necessary relationship between laws of physics is unknown, as is exactly how much variance would render a possible world impossible to live in.)
Therefore, if the meter is invariant over possible worlds, it isn't any of its definitions. All of them were efforts to specify a meter in ways we can actually use. This corresponds to attempts to find the values of physical constants: we assume that the value is constant (at least in this world) and employ various means to get better and better estimates. The speed of light has been measured in various ways over the centuries, in ways to get a better value for it. This hasn't changed the speed of light.
So, assuming that the kilogram is something that is the same in every possible world, the impending change from the mass of a particular artifact to a more physics-based measure doesn't affect it at all. It's just a better way to try to measure it.