Like most of these things it will probably end up depending upon the facts of the case, but it is an interesting one.
First thing that occurs to me is what "the public" would think of the police force after the accident. Not a strictly legal issue, but how would the force argue that what it had done was sensible.
Because it involves some sort of police work, I see no reason why a PAOC is needed per se. If a police force wanted to hire a helicopter from an AOC holder to, for example, do some aerial photography or search for someone, then as long as the helicopter was operated in acordance with the AOC and the normal ANO rules, I do not see a problem. Under an AOC, I have done work for police forces and other enforcers in the past.
If they just asked a nice kind PPL(H) who had a helicopter if he/she would take a nice policeman for a helicopter ride, again I can't see what rule would be broken either, but there could be some peripheral questions raised. It would be necessary to show that there was no "valuable cosideration given or promised", to quote Article 130. So no going easy on the pilot's next speeding offence!
I guess it may be OK as an exceptional event, but it surely cannot be sensible if regularly carried out. These days officer "health and safety" is a big issue. I am sure that someone like the Police Federation would ask for the risk assessment that justified using a private operation and pilot of unknown quality.
Flying Lawyer's thoughts would be interesting too.
All this would need to be done in accordance with all the standard low flying rules, cos none of the PAOC exemptions would apply.
I wonder what the aircraft insurance company would say though, or the police officer's life insurance company. I can imagine them trying to wriggle like hell after an accident to avoid having to pay out.
The other thing that occurs to me is a legal question about what difference it might make that you are not on the UK mainland - definitely don't know the answer to that one.
Email me if you want to make it less hypothetical!