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First, there is Upper German Butzerl (from Butz, eventually from Burkhard) as a hypochoristic name prevalent in the region and time in question.

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I am also thinking of the German word Putz "attire, adornement, ornament" (now archaic) but I have no idea if this relates to a Hebrew masculine name as a translation; a quick Wiktionary search reveals nothing relevant to me. The somewhat archaic German words Kopfputz "head wear" and sich aufputzen "to dress up" relate to this concept; today the normal meaning of putzen is "to clean, to make clean".

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Going deeper into Butz/Putz I recover another meaning, "mask", as in Fasnachtsbutzen "people wearing carnival masks".
Last, but not least there is the Butzemann "bogeyman", a mythological creature from German folklore recently becoming a meme in Catalonia.

Your understanding is mostly correct in terms of the word putzen meaning "cleaning" in modern German, but the medieval use of Putz (especially in Putzlein) carries a different meaning. In this context, Putzlein likely refers to a "small lump" or "small piece" based on its Middle High German origins, and over time, it became a name. This name could have been a nickname that eventually became hereditary.

Undoubtedly, thy cogitation upon the nomenclature Putzlein bespeaks a perspicacious foray into the philological substratum whence it derives. Indeed, in contemporary Teutonic parlance, putzen is oft connoted with ablution or embellishment, whilst Putz doth signify not solely ornamental facings but also the praxis of purification. Yet, if one delves into the arcana of Middle High German, Putz—erstwhile bereft of such modern embellishments—did bespeak "a lump," "fragment," or "stump," a semantic vestige whose vestiture hath since been obscured by linguistic vicissitudes.

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The appellative Putzlein, endowed with the diminutive -lein, appears an affectionate diminishment, perhaps imbricated with the vestigial denotations of materiality or corporeal modesty. Such semantic palimpsests, preserved within the Yiddish vernacular, oft evince a diachronic fidelity to Middle High German, a lineage unfractured by the truncations of modernity. Might thou discern other vestiges of such philological archaisms within analogous appellations?