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All of those answers are talking about the energy required to stop the moon dead in it's tracks.

What if we just wanted to slow it down 0.0000000000000000000000000000000000001% ? (which will alter it's orbit).

That seems totally doable.

We are already doing that. A orbital mechanics maneuver used in deep space spaceflight called a gravity assist or slinghshot trades a cosmic object's kinetic energy to change a spacecraft's flight path. Since that takes a large amount of energy at the speeds spacecraft go, an equal amount of energy must get subtracted from that object. Of course, the same amount of energy that's very significant for a spacecraft is meaninglessly small for an object such as a natural satellite like the Moon or a planet as Jupiter. So if you slingshot around Jupiter you are altering Jupiter's orbit around the sun an almost infinitesimally small amount, but you are altering it.

But of course, the question was asking for a significantly altered orbit, which this most definitely isn't.

Solar-powered railgun accelerating rocks to lunar escape velocity in directions chosen to make the desired delta-v. Speed the moon up or slow it down or change whatever other orbital elements you please.

This would work better to change the orbit of a small asteroid. For the moon it would take (a lot) longer.

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