The problem lies in how will you define the other parameters over here.

First up is the question of defining that 1 KG itself and the configuration; how will we define the parameters of the copper wire? (edit: to clarify after reemrevnivek's excellent comment the configuration of the solenoid is important because the strength of the magnetic field generated is directly related to the configuration of the coil) We can't use the radius, length of the copper wire accurately because as it gets heated up it will expand not only longitudinally but radially as well.

Second, how will the magnetic field be measured to the precision needed? If you use induction of current in another coil then according to lenz law that will create a downward force on our coil leading to inaccurate measurements (I'm assuming that it will be constantly measured) , this means that the coil needs to be strictly constrained, by how do you constrain it without adding/rubbing material on to the surface of the copper wire?

The magnetic field in a coil is dependent on the number of turns and current in a coil (as well as, to a lesser degree) the position of the wires in the coil. It is almost completely unrelated to the number of copper atoms in the coil.

Read up on the Biot-Savart law if you want to know more about calculating magnetic fields.

I guess you didn't read the article. That method is called the Watt-Balance method.