"Stopping Power" is the same for any decent brake, single or double pivot. All of them will lock both wheels of a bike and you can't get more than that. However, dual pivot brakes require less hand pressure for a given controlled stop or speed modulation and that can be very useful on long downhills where you have to apply the brakes frequently and for an extended time.
You should be able to find newer used double pivot Ultegra or 105 brakes, or even some earlier dual pivot Dura Ace brakes for well within your budget. One recommended upgrade is to fit what ever brake you get with Kool Stop Salmon pads.