See, this is what I'm concerned about. Common bullets that have plenty of legal application being caught in the snare of a restriction on AP ammo. Particularly when it would have little to no effect on officer deaths or the general safety of the public. The Hollywood shootout guys would have gotten the armor piercing rounds no matter what, as would any "serious" criminal (drug dealers, bank robbers, car jackers, etc.).
If you're wanting to prevent "casual" violent criminals from getting access to them, I have to ask, "What is the point?" First, such a casual criminal is likely going to handgun and as far as I know most states (at least those likely to enact this kind of ban) already have laws where a permit needs to be presented in order to buy the handgun ammo legally. It's far more likely they will buy an illegal handgun with a small supply of ammo (possibly whatever was in the gun when it was stolen), have no permit, and only be able to get ammo on the blackmarket when AP rounds will be available regardless of the laws. Second, the (labeled, purpose specific) AP rounds are higher priced rounds. That deters and limits their purchase right off the bat to those that either have a justifiable use and can only use them, or to those few permit holders who have a premeditated criminal intent who are either well off idiots who don't know better or know they really need them and would have gotten them on the blackmarket anyway. Third, if a permit holder is going to commit a premeditated crime with a handgun, they too are likely to buy an unregistered, illegal handgun to commit the crime so as to avoid anything tracing back to them. Again, they're on the blackmarket at that point and will have access to the illegal rounds no matter what.
So we're only going to be significantly impacting 2 relatively rare types of criminal: 1) A permit holder, using his own gun in a premeditated crime, who is anticipating and preparing and willing to make the leap to cop killer. 2) A permit holder, who has a legitimate use for the AP round who engages in a spur of the moment crime of passion using his own weapon and ammunition supply who will encounter a cop and be willing to kill him.
For this vanishingly small group of troublesome criminals we will enact a ban on AP rounds, open the interpretive door to politicians, lawyers and judges willing to engage in some "evolving standards" logic to broaden the scope of the ban to include bullets not reasonably intended to be covered, enable the gun control advocates and politicians to use the ban as a stepping point for the next ban and the next ban, and deny the use of these rounds for those few legitimate civilian purposes that exist.
I'm sorry, I think I've just talked myself out of supporting your proposal, Devil.

Once you get past the first blush reasonableness of the proposal, I think you're left with two things underlying its proposal by gun control groups and politicans. First, it is a "feel good" measure designed to sound nice on TV sound bites which, because of the phraseology and ignorance of guns and ammunition in the general public (and even here among those who have some level of experience) will be very difficult to oppose politically. Second, when the ban has no measurable effect on the number of shooting deaths of cops it will be used as evidence of the need to widen the ban to other types of ammo and guns. Alternatively, when criminals still use the AP rounds in a crime after the ban is passed, it will be used to justify a new round of dealer and distributor regulation, a renewed attempt to hold ammo and gun makers civilly and/or criminally liable, and more ammo bans.
I wouldn't put it past the politicians to use a "success" as evidence for more bans. Imagine a cop shot with a standard hunting round after the ban goes into effect. They'll get on TV and say, "The bullet used in this crime was not an AP round when it might otherwise have been, however this tragic death has brought into sharp relief the increased the use of far deadlier rounds, designed for maximum wound size and killing capability. We have to redouble our efforts by expanding the ammunition ban to include some more bullet types that are just as dangerous or more so than the AP rounds. For example frangible bullets and so called hollow point bullets, designed to create massive wound cavities and inflict maximum carnage in the victim, must be at the top of the list. The fight is not over, we must make our cities and states safe by ridding ourselves of these miniature tools of death designed for maximum killing capability." It isn't a stretch. They already have laws in place in some locations to restrict the purchase of certain "dangerous" types of rounds (typically handgun ammunition) or have brow beaten retailers into dropping gun and ammo sales entirely. And the usual suspects have been trying for years to ban frangible ammo.
Devil, you proposed this idea. I hope you find this request reasonable: Can you make the case that it will make a significant difference, can be worded to not impact these common types of ammunition, can be worded to not fall prey to "evolving standards" judicial interpretations as an end around on the common ammo types concern, and not be a momentum building stepping stone for more, less reasonable bans of other ammunition and/or guns?