Defining your fundamental unit of mass in terms of a single arbitrary physical object costs a lot of theoretical purity, though it might not cause a lot of problems in practice. You mention a potential loss of precision. But we should gain accuracy by defining it in terms of fundamental constants. When the object itself changes, do we simply have a standard that drifts more than the physical constants of the universe? Do textbook publishers need to update all examples and problems using micrograms because the unit drift at that scale became significant? Should anyone without access to a reference object not get a real, accurate value for the kilogram? Do we keep using the mass value we know the reference object had, or does our kilogram really change with the object? Do we need to update any equations with a constant in them involving mass in any unit or derived unit in the equation once a year for high precision applications?
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Defining your fundamental unit of mass in terms of a single arbitrary physical object costs a lot of theoretical purity, though it might not cause a lot of problems in practice. You mention a potential loss of precision. But we should gain accuracy by defining it in terms of fundamental constants. When the object itself changes, do we simply have a standard that drifts more than the physical constants of the universe? Do textbook publishers need to update all examples and problems using micrograms because the unit drift at that scale became significant? Should anyone without access to a reference object not get a real, accurate value for the kilogram? Do we keep using the mass value we know the reference object had, or does our kilogram really change with the object? Do we need to update any equations with a constant in them involving mass in any unit or derived unit in the equation once a year for high precision applications?