I don't actually know how much influence tidal forces have on the Earth's interior temperature. But here are some thoughts to consider.

The Moon very small compared to Earth and much less massive. To get a sense for that, compare the Moon's entire radius (~1740km) to just the radius of Earth's core alone (~3480km). Just our core alone is twice the radius of the whole Moon.

The Moon is made mostly of rock, like our mantle. Rock is very not-dense compared to metal. It's so not dense that the change in density from the deepest part of the mantle to the shallowest part of the outer core is MORE than the change in density from our atmosphere to the crust.

And then our mantle (rock) adds on another 84% of Earth's volume.

So while there is going to be some amount of tidal heating experienced by Earth on account of the Moon, I suspect it's likely not significant. Earth is just SO MUCH more massive than the Moon. (But hey, I could end up being wrong!)