The complications of calculating the moon's contributions to earth's heat budget are touched on in Emergence of a Habitable Planet (2007) by Zahnle, et.al. in Space Science Reviews, a paper trying to determine when in earth's history the planet's surface would become cool enough to support life. The portion of the paper that relates to tidal heating suggests that the early moon contributed substantially to the interior heat of the earth system for about 1.2 million years, but as the moon's orbit widened, the earth cooled and solidified, and water condensed into oceans, the moon's contributions gradually became minimal.
Viscous damping of tidal motions generates heat. Therefore tidal
heating occurs most strongly in materials that are solid but close to
melting. This introduces the possibility of a governing feedback that
works through the dependence of viscosity on temperature. If tidal
dissipation exceeds what the atmosphere can radiate, the excess heat
raises the temperature, which lowers the viscosity, which in turn
lowers the rate of tidal dissipation. This looks like a stable
feedback. It follows that, while tidal dissipation was important, the
base of the mantle was solid but the rest of it was fluid, and tidal
heating generated almost all of the thermal energy radiated to space.
In the limit of an asymptotically thick steam atmosphere, tidal
dissipation would have been regulated to generating heat at the
runaway greenhouse limit of ∼140 W/m2.
In a caption for a figure the authors also note:
Tidal heating plays an important role in prolonging the magma ocean.
Tidal forcing wanes as the Moon evolves away from the Earth.
Thereafter heat flow is controlled by convection of the solid mantle.
By 4.4 Ga the global average heat flow would have been ∼0.5 W/m2.
Later in the Hadean typical heat flows would have been 0.2–0.3 W/m2,
not enormously larger than what they are now. For comparison heat flow
today is 0.065 W/m2 through the continents and 0.1 W/m2 through the