“Bowstring” is approximately the 67,700th most common word in the Gutenberg corpus, “bow-string” is not found in the 100K!  Each is used once, and in very different settings.  I am intrigued that Bilbo’s word that he thinks of internally to indicate his state of anxiety is not hyphenated, while the name for the tool that Bard used is.  Not a Shire word, then, yet expressed as a specialty word that Tolkien (translator) had no word for – different from a bowstring.  Perhaps the greater bow of a mighty man had a different sort of string.

OED does give “bowstring” the head word and “bow-string” in the examples.

• 05.127 Bilbo could see or feel that he was tense as a bowstring,
• 14.021 Then Bard drew his bow-string to his ear.

“bow-string | bowstring, n.” OED Online. Oxford University Press, June 2017. Web. 5 September 2017.