I'm not a physicist, but it's irritating to me that this article is using weight and mass interchangeably. I doubt they'd ever allow this, but the "simple" solution is to bring the weight aboard the ISS and capture its mass on an inertial balance.
Nobody can build a kilogram-scale inertial balance of sufficient absolute precision. If we knew how, we'd be doing it.
Also, the ISS is a tricky place to work for a precision measurement. It's electrically, seismically, and gravitationally noisy (and huge gradients). Precision gravitational measurements are generally carried out in dedicated spacecraft with careful attention to those concerns, if they can't be done on the ground.
Most people don't realize that the gravity on ISS is almost as strong as it is on the surface of the Earth. They're continuously falling toward Earth, going just fast enough to miss the ground.
