If you'll ask any average American who happened not to have majored in English or literature in college, you'll discover that the final syllables of "caribou" and "milieu" do rhyme, though imperfectly. Ogden Nash might have gotten some mileage out of them had he thought to do so.

So far as I know, the case for Coward's authorship rests on these facts only:

Both Noel Coward and "Eskimo Nell" were extremely clever and perverse, both were in circulation in 1931-32, and both used "caribou."

To me that's not a case; it's a suspicion. Other clever and perverse people were in circulation at that time, though most of them presumably kept lower profiles than Noel Coward.

Short of a signed and notarized confession, I'd have to see more actual evidence before concluding that Coward was the poet in question.