Let us settle this with Strong's Hebrew Dictionary - the standard dictionary for translating the Bible
Lev. 11:13, 19 And these are they which ye shall have in abomination among the fowls*...And the stork, the heron after her kind, and the lapwing, and the bat.
*Fowl is a translation of ofe. Its etymology is from ?ûph (pron - oof; a bird) = as covered with feathers, or rather as covering with wings, often collective: - bird, that flieth, flying, fowl.
Deut. 14:11 Of all clean birds* ye shall eat.
Deut. 14:18 And the stork, and the heron after her kind, and the lapwing, and the bat.
a little bird (as hopping): - bird, fowl, sparrow.
From H6852 (Etym. tsâphar tsaw-far' A primitive root; to skip about, that is, return: - depart early.)
So we see that in Leviticus, although a bat could be in the category of birds OR winged creatures (nothing to do with vertebrae), the choice is resolved in Deuteronomy, where the bat is clearly a bird.
God said it; I believe it. Case closed. : )