As you say, swapping the order of adverb and verb is a poetic construction -- but that doesn't always make it good poetry. There's nothing grammatically wrong with "nothing know", but in my opinion it does not flow elegantly -- or, as you might poetically say
It does not elegantly flow.
Unfortunately, it's difficult to define the rules (such as they are) around what sounds good and what does not. As in any language, this usually relates to things like rhyme and rhythm; that is, whether words have complementary sounds, and what information the natural intonation and word order adds to the sentence.
For example, there is a natural emphasis on whatever is the last word or phrase in a sentence. Consider the difference between these:
I don't know who killed him, but I do know Sam didn't do it.
I don't know who killed him, but I do know it was not Sam.
In the second sentence, "not Sam" makes the statement more emphatic, as if the speaker is more certain about their belief.
In a similar way in your example, "I know nothing" emphasizes nothing, as that creates a contrast with the earlier "know". Reverse the order of the words and instead you've emphasized the verb "know", removing the contrast and much of the significance.
However, in a different context, this parallel might be exactly what you want to say:
We only really know what we believe that we know. Unless we are comfortable ignoring uncertainty, we could never even get out of bed for fear of falling through the floor.