The (no matter just how great) translator gave you a subpar explanation.
The tenor of this sentence (regardless of its belonging to a literary piece) is conversational, and as such, the sentence happens to be part of English. But if the author intended it (he didn't) to be grammatically cracking, he failed. The error is not to be found in the predicate, though.
First, the predicate kick into is correctly used in the perfect instead of the simple aspect. For the reality of the story, the effect of the momentary action of kicking the adversary's gun is definitely not momentary, because after being disarmed, the antagonist becomes incapable of ending the protagonist's life! That's one of the raisons d'etre of the perfect aspect: to covey the lasting importance of the action's effect. It's not only that I kicked his gun into whatever; I have kicked it into whatever and now the effect of that is most definitely felt: I can now much more easily fight for survival.
Second, the past tense has correctly been chosen over the present tense, because the narrator is recounting events, not reporting them as they unfold. Thus,
tense + aspect —>
past + perfect —> Past Pefect. So far so good.
But thirdmost, the temporal adverbial phrase "the next instant" is a momentary one. You cannot have a time-effect of an action of a perfect verb squeezed into merely a point on a timeline. To have a grammatically valid sentence, the time of that adjunct should be lengthened, however little or much, to coincide with the predicate:
But before he could shoot, I knocked it out of his hand, and by the next instant had kicked it clean over the edge of the rock into the sea.
In this case, the by doesn't lengthen the time much: the action starts at a certain instant and already by the next instant, the effect is "had." Still, the time is no longer momentary; we at least now have two instants. That can no longer be represented by a single point on a timeline.
Again, that'd be a Standard English sentence, but for a literary work, Fletcher's sentence is perfectly standard.
I have just now read Stoney's comment, so allow me this cop-out:
The adverbial phrase "the next instant" might also be understood as a radical ellipsis of "the situation in the next instant was such that," which wouldn't require a by, so the sentence might, after all, be perfectly grammatical.