Wrong. For rotation you don't look at mass, but at moment of inertia. That scales like ~M*R^2 or ~rho*R^5 for a density of rho and distance R from the axis of rotation. So even if the core is much denser, it contributes a lot less to the earth's total moment of inertia. Earth's rotation is even measurably affected by plants changing throughout the seasons. Glacial melts and earthquakes also have measurable effects that need to be compensated for. See here for the plot:
That second paper summarises that the changes are quite probably due to deformation which is caused by core movement.
We are not debating seasonal variation.
Mass scales like density times radius cubed. The first M would technically be a function of position. So you would integrate over slices of varying M not just because of varying density.
