The verb G. biltha- “flutter, flit” appeared in the Gnomish Lexicon of the 1910s, an elaboration of G. bil “bird” (GL/23).
Neo-Sindarin: Since G. bil “bird” is no longer used, for purposes of Neo-Sindarin I would alter this verb to ᴺS. gwilwa- “to flutter, flit”, inspired by Q. wilwa “fluttering to and fro”.
A word for “bird” in the Gnomish Lexicon of the 1910s appearing next to G. bilinc “sparrow”, but this word was deleted and the gloss for bilinc was expanded to “a small bird, especially sparrow” (GL/22-23). The form bil appeared in a couple other places in the Gnomish Lexicon (GL/23, 31), but seems to represent a “root” rather than a word. Tolkien indicated bil was derived from ᴱ✶du̯il (GL/31), but the exact mechanism whereby du̯- became b- isn’t clear, but a similar change is seen in 1920s ᴱ✶du̯ag- > ᴱT. baga- “beat” and ᴱ✶tu̯ak- > ᴱQ. pak- “apply, attach” (PE14/66).