An Etymological Dictionary of the German Language/Annotated/Bausch
Bausch, masculine, ‘pad, bolster,’ from Middle High German bûsch, masculine, ‘cudgel, blow causing blisters, swelling.’ If ‘cudgel’ is the primary sense, the word may be connected with Middle High German bôȥen, Old High German bôȥȥan, from bautan (see Amboß, Beutel, Beifuß); bût- would be another stage in gradation, and before the suffix sch from sk the dental would inevitably disappear; compare Latin fustis, ‘cudgel,’ from *bhûd-stis.