Interesting problem. And back to what I asked earlier, about proportion. "What percentage or proportion of the total energy in the Earth-Moon system would be heat energy?"
This is purely a guess, so please excuse. Given the masses of Earth and Moon, and excluding other influences (Sun et al) I'm thinking that the total energy between the two would be some hundreds times more than that given over to friction.
Having been a mechanical engineer in my distant past, I'm looking at it as being something like the efficiency of two well oiled gears or cogs, with one difference being that gravity is like energy normally delivered through the shaft (which I guess is where the analogy breaks down), but in engineering terms, I would expect something like 98% efficiency. And with celestial bodies suffering no physical friction or atmospheric friction, I would expect something like 99.99% efficiency.
Am I looking at this right?