Even though there is no experimental evidence, that the Alcubierre drive works. The Einstein field equations permit this solution. For Gravitoelectromagnetism there are hints and ongoing efforts to test those hypotheses. See the up to date wikipedia articles on the Alcubierre Drive and Gravitoelectromagnetism. But whether there will or won't be experimental evidence does not matter for my question. My question is: Are the approximate Gravitoelectromagnetism equations sufficient to explain the Alcubierre Drive?
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It isn't obvious to me why there should be any relationship between the Alcubierre Drive and gravitoelectromagnetism. The Alcubierre drive is a valid solution provided you can lay your hands on some exotic matter, but then if you have a stock of exotic matter to hand all sorts of problematic solutions to GR are possible. The conclusion that most of us draw is that while exotic stress-energy tensors can be written down, they don't correspond to physical reality. It's like putting a negative mass into Newton's equations - it can be done, but you won't get any meaningful output. Anyhow, gravitomagnetism isn't required.