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For two graphs and , write $G \stackrel{\mathrm{rbw}}{\longrightarrow} H$ if has the property that every proper coloring of its edges yields a rainbow copy of . We study the thresholds for such so-called anti-Ramsey properties in randomly perturbed dense graphs, which are unions of the form , where is an -vertex graph with edge-density at least , and is a constant that does not depend on . Our results in this paper, combined with our results in a companion paper, determine the threshold for the property $G \cup \mathbb{G}(n,p) \stackrel{\mathrm{rbw}}{\longrightarrow} K_s$ for every . In this paper, we show that for the threshold is ; in fact, our -statement is a supersaturation result. This turns out to (almost) be the threshold for as well, but for every , the threshold is lower; see our companion paper for more details. Also in this paper, we determine that the threshold for the property $G \cup …