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。
so, i would suggest the etymology of "完美" is derived from "完整" + "美好"
完(adverbial - entirely / wholly) + 美(adjective - beautiful / excellent)
完美 is a compound word for "perfect, perfectly, perfection"
|wan2 | pinyin
 [v] complete; finish; bring to a conclusion
 [v] run out; use up
 a Chinese family name
 beautiful; pretty; pleasing; fine; fair
 good; excellent; exquisite; nice
 be pleased with oneself
 [v] praise
|jyun4 mei5| jyutping
|wan2 mei3| pinyin
perfect, perfection, perfectly