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the 2 body problem of earth and moon is suppose to be perfectly described by Newtonian mechanics / Kepler's laws.

How ever the moon's orbit is been moving away from earth in contradiction to the orbit described by Newtonian mechanics.

So are we to conclude that even Newtonian mechanics does not perfectly describe the solution to 2 body problem?

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No. The moon and earth are not an isolated two body system.

Specifically the reason the moon is getting further away is because of tidal friction - angular momentum is being taken from the Earth's rotation and imparted to the moon's orbit. There is obviously an upper limit to how much this continue (namely, the limit of $L_{\rm moon} + L_{\rm earth}$ with respect to their center of mass).

This is all well understood within the framework of Newtonian mechanics.

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    $\begingroup$ Note that the corrections to the Moon's Newtonian motion due to general relativity are of order ten meters and have been measured at the centimeter level. $\endgroup$ – rob Jun 25 '17 at 4:06

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