Given that road bikes are generally faster and lighter I wonder why police, much like their use of HDs, choose to use mtbs? The only real advantage a mtb has for the most part is adeptly handling rough terrain. Road bikes accelerate faster being that their usually less than half the weight of a mtb and SS's are nearly silent in operation which in tactical operations would be a distinct advantage. Although the harsh ride of the rigid frame would make a full (8hr) shift very fatiguing.
I think you should test your idea and let a bike cop see you take a bag from a little old lady on your road bike in the environment a cop bike would be present.
Or make friends with some bike cops and have a friendly challenge that doesn't put you in the klink.