I would suggest that the surge you feel is not the prop "really biting" as the efficiency of a constant speed prop plots as a fairly constant curve. ie: no airspeed/rpm/etc regimes where it is dramatically more efficient at producing thrust. Additionally, the prop installed on a stock 182 is most leikely going to be one which is most efficient in the cruise regime (vs the aftermarket seaplane prop on my C-180 which is optimized for low speed thrust)
Rather the difference in push you feel is due to the fact that the constant speed prop allows the engine to turn up to max RPM (and thus produce max power) at low airspeeds, vs the fixed pitch prop where at low airspeeds the engine RPM is necessarily low, and thus power output is also well below max. A fixed pitch prop airplane will not produce max rated power until such time as the airspeed is high enough to allow the RPM to increase to redline. Unless you have a very flat pitch prop, that probably isn't going to happen on the takeoff roll.