Shouldn't all moon orbits be inherently unstable?
The answer to your simple question is that while a circular orbit could remain in the absence of any other gravitational fields, in reality you'll have an ellipse and it can be stable from the beginning, remember stable has a wide range of values.

If you allow drag through tidal forces etc then the orbit will change, eg our Moon is slowly getting further away from us, but we still consider this stable over realistic timeframes.

Additionally, the actual orbit will track out a sort of helix as the planet travels round the sun.