I have one, though it has a different set of birds than yours, and for me the clock will keep going while the birds stop chirping after a while. Mine takes three batteries - one by itself and two some space away. I'm guessing the one by itself is the one for the sound, but could be wrong. Anyway, the clock part keeps going long after the birds stop.
Adding to the problem, my owl chirps and some little bird hoots, and my fiddling with it has not managed to rearrange this. Luckily, I think it's funny.