SeanM, one point in favor of your view. In 1580 several printings appeared, and the authors (or publishers) quarreled over it and at least one parody was produced. It is likely that there were many more printings that were not preserved.
It certainly does get stuck in the mind. Was it, as you suggest, psychological warfare? Perhaps originated by a fiendish Spanish mind-twister and introduced to the English to addle their brains?
Good story, but there seemed to be a lot of other sonnets or whatever of similar vein at the time. Luckily, these aren't remembered. They may have been the 16th century equivalents of the mind-numbing Harlequin paperbacks of today.