These notes put some numbers on @ACuriousMind 's answer: one needs to be looking at length scales of 100 Mpc and greater for the FLRW metric to be a realistic description of reality. So one cannot expect the scalefactor expansion of spacetime to apply to our Earth-Moon system because it doesn't fulfill the assumptions that justify the FLRW metric.
Perhaps a more addressable variant to your question would be to ask what difference does a positive cosmological constant make to a metric that does describe the Earth-Moon system. This is a question that can be answered, and it is reasonably straightforward to go through the derivation of the Scwharzschild metric but with a positive cosmological constant. One finds that the metric changes as follows:
$$g_{t\,t} =c^2\left( 1-\frac{r_s}{r} -\frac{r^2\,\Lambda}{3}\right)$$ $$g_{r\,r} = \frac{c^2}{g_{t\,t}}$$
and the cosmological constant, if small enough, does not change the basic character of geodesics; it will however shift the radiusses of stable orbits. These notes sketch how to work through the computation; the radially symmetric system with nonzero $\Lambda$ being Problem 23. in Chapter 23 of Moore, Thomas A., "A General Relativity Workbook".
So there is no ongoing spacetime expansion in this system: orbits are just a little bigger than they would be with $\Lambda=0$ and some weakly bound orbits would become unbound with positive $\Lambda$. Therefore, we would not expect the Moon to be drifting away any faster that it is owing to nonrelativistic effects.