The target has clearly, unquestionably identified himself as a target that needs shooting. The only issue is where he is; the instant he comes thru the door I'll know that important tidbit and act accordingly. Shooting thru the door increases your chance of missing...it may give him second thoughts, you may waste the shot, he may have a hostage that you might plug instead, etc.
Note that the door will
(a) obscure your view of the target
Consider the angles. He's 1-2 feet behind the door (he's kicking it, remember). Don't know about you, but I'll up against the wall the door is on...to fire well I'd either have to shoot thru the wall at a sharp angle to hit behind the door (even then, not much will get thru closet door, closet wall, and hall wall), or move in front of the door (just as the door comes crashing down and putting me right where Mr. Perp expects to see me).
Also not likely to get a center-of-mass hit when I can't see the mass.
(b) absorb shot energy
While a shotgun slug may not notice my door, a 9mm hollowpoint may lose much energy.
Take the extra seconds to gain composure, acquire a strong stance, get behind cover, and prepare to plug the target once you know where it is (i.e.: comes thru the door).
Cooper's Rules always apply. Such questions - and they are valid - are best answered within the context of The Rules.
What if I do shoot thru the door? I'll assume a shotgun w/slugs. I know the target: anyone killing my dog in my house at oh-dark-hundred IS a target. I know the location: to kick the door requires target placement within a small area (about 4 square feet). I know what's behind it: more house, a berm, and beyond those mostly empty space as far as one can see; I live alone, so the likelyhood of innocents present is nil. I've identified the target and what's behind it, though not quite to the degree of waiting for the door to crash open. Rule #4 is still followed...but that's because of the known situation; this may not work for others with in-house family or close neighbors.
Settle in, get ready, and pop the intruder as he comes thru the door; a better implementation of Rule #4.
Don't violate the rules. Be aware of how they apply to your particular situation. Shooting thru the door might work (and follow Rule #4) for a rural bachelor, but not work (and violate Rule #4) for a suburban father.