The spin axes of the mantle, fluid core and solid inner core of the Moon precess at frequency $\Omega_p=2\pi/18.6$ yr$^{-1}$ though with different orientations, leading to viscous friction at the core-mantle boundary (CMB) and inner core boundary (ICB). Here, we use a rotational model of the Moon with a range of inner core and outer core radii to investigate the relative importance of viscous dissipation at the CMB and ICB, and to show how this dissipation is connected to the phase lead angle ($\phi_p$) of the mantle ahead of its Cassini state. We show that when the inner core radius is $>80$ km and the free inner core nutation frequency $\Omega_{ficn}$ approaches $\Omega_p$, viscous dissipation at the ICB can be comparable to that at the CMB, and in the most extreme cases exceed it by as much as a factor 10. If so, the viscous dissipation in the lunar core projected back in time depends on how $\Omega_{ficn}$ has evolved relative to $\Omega_p$. We further show that constraints on the CMB and ICB radii of the lunar core can in principle be extracted by matching the observed phase lead of $\phi_p=0.27$ arcsec; this requires an improved estimate of tidal dissipation and an accurate model of the turbulent viscous torque. Lastly, when our rotational model is constrained to match $\phi_p=0.27$ arcsec, our results suggest that the viscous dissipation at the ICB is likely insufficient to have ever been above the threshold to power a thermally driven dynamo.