This is taking too far the concept of relativistic mass. Einstein himself was not fond of this concept, according to his quote on this wikipedia article.
If I understand your question correctly, you find an apparent paradox pursuing the consequences of plugging the relativistic correction factors ($\gamma$) in both sides of Kepler's third law. On one side, one should measure the increase of the period because of time dilation, but on the other side, one should find a decrease in the period because the relativistic mass correction of the earth moving respect to you. You ignore the consequences of a possible Lorentz contraction of the orbit's size, but that is not important since none of Kepler's laws is consistent with special relativity or with the adequate corrections of gravitational fields when measured from moving observers.
For you, moving at $0.8c$, the moon completes an orbit in 45 days indeed. It is like a giant clock ticking slower just because of Lorentz time dilation.
The enormous gain of earth's kinetic energy respect to you does indeed modify its gravitational field, but in a very different way compared to just increasing the mass of the earth by a factor of $\gamma$. In fact, the gravitational field changes in just the adequate way for you to observe the moon's orbit modified according to the predictions of special relativity. For example, if you move in the plane of the orbit towards the earth, you will see the orbit of the moon (originally a circle, say) becomes an ellipse but with earth in the center, not in any of the foci. This is only to illustrate that gravitational fields, from moving observers, do not behave as a normal Newtonian field with an increased mass.