AIS has no minimum range , its a gps linked transmitter , AIS receivers ( commercial ones ) link directly into the radar ( provided its an ARPA type ) as well as directly into electronic chart plotters. Warships probably don't use electronic charts but considering the amount of radars warships use it does seem very unlikely that they wouldn't have had anti-collision warnings.
The big problem with AIS lies in the fact you can put a laptop between the GPS receiver and the transmitter unit and you can put in a quite sizeable offset ( have heard up to 10 miles ) , this is clearly illegal but the software is freely available and a cheap laptop with an interface card means it can be done ( has been done in the UK , lets just say the guy won't be doing it again... MCA.... ). As for changing what name comes up , yes you can do that BUT you can't change the IMO number which means using any of the online AIS websites will show the true ID of the ship which is why the worlds Navies ( and somewhat illegally as they don't have a dispensation from the IMO organisation or MCA the Scottish Fishery Protection ships , they also switch off ALL nav lights and darken ship again without seemingly legal clearance ... ) simply switch them off all together.