Trying to get the laser pointed at it would be a daunting task. A bird hit would result in one plasma-ized bird.
Question 'bout the bird : airframes are notoriously flimsy. Would the 'shrapnel' from peeling of the missile's skin upon impact with the bird affect targeting, or is the missile pretty much terminal at that point and so the angular deviation from the programmed course will be minimal?
Conservation of momentum, conservation of energy with the *bird* means the bird itself won't alter the trajectory: but the sudden discontinuity in airflow around what is left of the frame *might*.
NASA and/or the DoD used to do equation of state / CFD for hypervelocity penetrators, but that was for when they hit the target; and the airlines have done LS-DYNA birdstrike simulations on the engines; don't know if anyone's put the two together...
Cheers!