Welcome to the wonderful world of animation
The up and down motion seems to be out of sync with the wingflapping.
Wings flapping down = driving the body upwards. At all other times there is a constant gravity pull, and perhaps even an extra downwards drive when the wings are going back up.
I tried to fix that a bit, but I'm sure it could be done even better:
Also there is some funny stuff going around in the head and the neck. It is nice for the head to move separately, but that the size/height of the head and neck change is a bit strange...
As for bat anatomy: he seems to have two wrists! remove the one on the far end! Also his wrist would be more extended in the downflapping part, and his elbow more bend during the upwards recoil.
A quick google led me to
this ref pic, which seems fairly accurate to me. That page also links to one of my favorite animation articles of all time:
How to make birds fly goodGood luck with your animation progress!