Grammar tools are notoriously prone to getting subtle distinctions wrong. They apply some general guide, such as "avoid double negatives" and make of it an absolute ruke, and flag anythign that looks like a double negative, ignoring both things that look like double negatives but are not, and things that are, but in the particular circumstances are perfectly acceptable. Such tools can be useful to point out oversights, but if a tool points out a sentence, phrase, or word that seems acceptable, feel free to override the tool. (Or check with this site if one has time).
The given sentence,
May it not become nothing, like the clouds that pour rains and then disappear
is not a case of a double negative in the sense that grammar books often frown on. It contains two negative words in the same clause, but they are not being applied to the same thing. The word "not" is used to modify the verb form "become" while "nothing" is the direct object of "become". As the answer by Mohsen from Iran says, this clause could be rendered as "May it not turn into nothingness".
However, both the original sentence and the suggested variant strike me as trying too hard for a poetic or old-fashioned flavor. I might, however, change this view if I saw more context for the sentence. In any case that is a matter of style, not grammar.