Here is the etymology of the word "bat":
flying mammal (order Chiroptera), 1570s, a dialectal alteration of Middle English bakke (early 14c.), which is probably related to Old Swedish natbakka, Old Danish nathbakkæ "night bat," and Old Norse leðrblaka "leather flapper," so original sense is likely "flapper." The shift from -k- to -t- may have come through confusion of bakke with Latin blatta "moth, nocturnal insect."
As you see, there is nothing in common between two words, they just sound a bit similar (and as you see above the word "bat" could even be something like "bak" today. I'm Bakman!)