The word ᴱQ. qildare “bat” appeared in both the Qenya Lexicon and Poetic and Mythological Words of Eldarissa of the 1910s as an elaboration of ᴱQ. qilda “quiet” (QL/78; PME/78).
Neo-Quenya: I’d retain ᴺQ. quildarë “bat” for purposes of Neo-Sindarin, still based on ᴺQ. quilda “quiet”.
A root in The Etymologies of the 1930s glossed “tread”, with the derivatives in both (Middle) Quenya and Noldorin having to do with walking, trampling and paths (Ety/BAT). It may have been replaced by later root √PAT with various path and walking derivatives (PE17/34), or the two might have coexisted as light (√PAT) and heavy (√BAT) variants.