What am I overlooking?
When in doubt, blame it on global warming.
They DID foresee that weight can vary. That's why they aren't measuring the weight of the masses. They are measuring the difference in their masses. They are using a highly accurate device equivalent to a double pan balance. If you place equal masses on each pan of a double pan balance, the needle points straight up. If one is more massive, the needle tilts to that side.
If you took equal masses to the moon and checked them on a double pan balance, they would be shown to be equal, even thought the gravitational field strength is different.
As several people on this thread have mentioned, the varying gravitational field is irrelevant in this comparison.