No wonder that even in France the wheat from Poland was known under the English trade name of "Polish". But the French have always been very creative in butchering the words they loaned from English (mind you, the French word for "shampoo" is "''shampooing''" and the French for "walkie-talkie" is "''talkie-walkie''"), so it shouldn't come as a surprise that they would write the English word "Polish" as "''Polisch''" or even "''Poolish''".
And now its it's time for my own proposition: the poolish method was developed in France, but it gave the best results with the "nervous" Polish wheat, which was known in France as "Poolish" (among other misspellings) and lent its name to the method. Only once the name "poolish" had caught on, along with the pseudo-English pronunciation to match the butchered spelling, was the French phonetic spelling "''pouliche''" (which also happens to be the French word of "filly") coined.
Is this proposition supported by any historical sources? No. Is it far-fetched? Definitely. But is it more of a stretch than the other etymologies presented above? I shall leave it for the Reader to decide. So what do we know for sure? That poolish was invented in France rather than in Poland, and in the second half of the 19th century, so it couldn't have been popularised by August Zang – let alone by Marie Antoinette. But the origin of its name will remain a mystery.