You mean moment of inertia, not angular momentum.
You could measure all of them! Given the moments for the three principal axes at any point, you can use the parallel axis theorem to calculate all the rest. In general, there are 10 degrees of freedom: 3 for the position of the center of mass, 3 for the axes, three for the moments, and for the the total mass.
For a nicer way to count and to do the math, you have the inertia tensor at the CM (a 3-dimensional rank-2 symmetric tensor, 6 DoF) plus the location of the CM and the mass.
In any event, this is a cute tampering-detection trick, but I would have started with an X-ray or CT scan.