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For two graphs G and H, write G⟶ rbw H if G has the property that every proper colouring of its edges yields a rainbow copy of H. We study the thresholds for such so-called anti-Ramsey properties in randomly perturbed dense graphs, which are unions of the form G∪ G (n, p), where G is an n-vertex graph with edge-density at least d> 0, and d is independent of n. In a companion paper, we proved that the threshold for the property G∪ G (n, p)⟶ rbw K ℓ is n− 1/m 2 (K ℓ/2), whenever ℓ≥ 9. For smaller ℓ, the thresholds behave more erratically, and for 4≤ ℓ≤ 7 they deviate downwards significantly from the aforementioned aesthetic form capturing the thresholds for large cliques. In particular, we show that the thresholds for ℓ∈{4, 5, 7} are n− 5/4, n− 1, and n− 7/15, respectively. For ℓ∈{6, 8} we determine the threshold up to a (1+ o (1))-factor in the exponent: they are n−(2/3+ o (1)) and n−(2/5+ o (1)), respectively. For ℓ …