In Constrained Correlation Clustering, the goal is to cluster a complete signed graph in a way that minimizes the number of negative edges inside clusters plus the number of positive edges between clusters, while respecting hard constraints on how to cluster certain friendly or hostile node pairs. Fischer et al. [FKKT25a] recently developed a $\tilde{O}(n^3)$-time 16-approximation algorithm for this problem. We settle an open question posed by these authors by designing an algorithm that is equally fast but brings the approximation factor down to $(3+\varepsilon)$ for arbitrary constant $\varepsilon > 0$. Although several new algorithmic steps are needed to obtain our improved approximation, our approach maintains many advantages in terms of simplicity. In particular, it relies mainly on rounding a (new) covering linear program, which can be approximated quickly and combinatorially. Furthermore, the rounding step amounts to applying the very familiar Pivot algorithm to an auxiliary graph. Finally, we develop much simpler algorithms for instances that involve only friendly or only hostile constraints.