Would it be possible to define the KG as the amount of copper wire, when arranged in a configuration (a coil?), that causes a magnetic field of a certain size when a specified current is applied?
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?
Read up on the Biot-Savart law if you want to know more about calculating magnetic fields.