Shoki shamisen no kenkyū 初期三味線の研究 ("Research into the early shamisen") is a recent collection of papers by Japanese music historian Gamō Satoaki 蒲生郷昭. The first part of the book, which I read this week, is an exhaustive philological analysis of the early history of the word shamisen, beginning with its well-known first appearance in the late 16th-century diary of Uwai Kakuken/Satokane (上井覚兼日記). (Spoiler: "しやひせん", in kana, twice.)

Gamō concludes from his investigations that the word shamisen entered Japanese from a Ryūkyūan language (rather than from Chinese directly), pointing out that most early attestations were in kana rather than kanji, even in otherwise kanji-heavy works; this suggests that these early witnesses either had no Sino-Japanese morphological model of the word or wanted to emphasize its exoticness.

Following the early kana phase (Gamō goes on), several different kanji spellings (ateji) are seen. The current standard, 三味線 (borrowing the kanji for sanmai 三昧, a loan via Chinese of Sanskrit samādhi), was first adopted by Kyoto-based haikai poets in the Kan'ei era (1624-1643). However, this spelling didn't catch on outside haikai circles until the Manji era (1658-1661), when Asai Ryōi 浅井了意 began using it in his popular kanazōshi works:

... 三味線を引寄せ、でつるてんと引く撥音、軈て買手をあがり鯰にせんといへる響きあるゆゆしき ...

... she pulls the shamisen to her, and grim indeed is the way the de-tsuru-ten sound of the pick on the strings seems to sing of her patron's impending penury ...

This, it seems, was the final push that made 三味線 the winner.

Incidentally, in Gamō's discussion of the word's appearance in the 1595 Dictionarium Latino Lusitanicum, ac Iaponicum, he throws out a mystery I can actually solve. For context, here's the original DLLI entry in full:

Lyra, ae. Lus. Viola, ou outro inſtrumẽto de cordas. Iap. Fiqu biua, xamixen. ¶ Itẽ, Hum ſino caleſte. Iap. Foxino yadori.

Having covered the directly relevant first half of the above, Gamō adds, "After this in the same entry, following some other Portuguese, it says 'Foxino yadori.' I suppose this is 'Starry abode,' but what can it mean?" (なお、同項ではこのあとに、別のポルトガル語につづけて "Foxino yadori" とある。「星の宿り」であろうが、何を意味するのだろうか。) Well, it refers, of course, to the Lyra, the constellation. Mystery solved.

Also, note that the first Japanese translation for lyra above is fiqu biua, "biwa that you play [by plucking]". Presumably this is to avoid confusion with the other biwa, the loquat. A brief glimpse into an alternate reality where kanji were eliminated entirely from Japanese orthography, and a thousand "ink pen"-style disambiguatory elaborations arose in response.