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. That's a staggering distance, and equivalent to timescales amounting to the whole Mesozoic era, comprising the rise and fall of the Dinosaurs! So one cannot expect the scalefactor expansion of spacetime to apply to our Earth-Moon system simply because the system doesn't fulfill the assumptions that justify the FLRW metric.
Perhaps a more addressable variant to your question would be to ask what difference a positive cosmological constant makes 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.