Althought I have never read a bible, I can say that it is a relative clause but the subject has been omitted:
"that" can refer to a heavenly creature or an angel or maybe another bird
so "which hath wings" becomes a relative clause beginnig by "which" and that describe or modify "that"
it's like saying :He who has enough power, shall win.
But I know that the idiom "a little bird told me" is used when you want to hide the name of the person who gave you an information or a secret.
I hope that this will help you.
Student or Learner