Where does the word 完美 come from? I suspect it comes from "又做完了，又做得美丽," since that would make some sense. Is it something along these lines?
there's an earlier usage of "完美" in 唐 dynasty. in 漢文大藏經﹒毗尼作持續釋﹒卷五:

釋　若犯妄語。譬如大石破為二分終不可還合者。謂石大　完美　。四覩奇觀。比丘戒淨。人天欽仰。若犯妄語波羅夷法。不得還成比丘梵行。布薩羯磨不共眾僧。如石兩分。破壞本質。失眾敬瞻。棄置不盻故。

http://tripitaka.cbeta.org/mobile/index.php?index=X41n0730_005

so, i would suggest the etymology of "完美" is derived from "完整" + "美好"