The story
From bat + -let. Probably a spurious word, in the 20th century reborrowed from word-lists. Both this and batler are only known from the same Shakespeare locus; neither is it known that battler means a fuller’s beetle but him who beetles or “posses” the clothes. However for the meaning of a flat cuboid on a handle to clean textiles by muscles battril, which could be a metathesis of batler, is known to have been used in the Lancashire dialect, such as by Tim Bobbin on multiple occasions.
Etymology from Wiktionary, CC BY-SA 4.0.