If you take a piece of string to measure the radius, and then later c changes and you measure it again, the two pieces of string will have a different length when put next to each other. Saying it doesn't change because the length of the meter changed in the meantime is playing with words. The reason humans came up with a definition based on light is the assumption that c is constant and thus would be a more reliable choice as a standard than some physical bar of metal kept in Paris. But if the constant is not constant after all then something else will be needed as reference.