Laser beam diverges. Beam energy density on the target is much less than on the laser mirror unless the target is very close, so the heavy cooling on the target might not be necessary. And as for increased dwell time - these lasers work in bursts on the order of one second, maybe two or three.
The burst mode makes more sense if you are looking to maximize 'shock kill'. Physical affects are probably slightly similar to a supercavitating underwater torpedo traveling through an underwater thermal break. Temperature and pressure differentials. Yes the beam energy at the primary mirror is going to be greater then the applied target beam energy. The main question would be, how much energy loss per distance, and that my friend is probably heavily classified. Would also be affected by atmospheric conditions.