In this paper, a class of tight Bayesian bounds on the mean-squared-error is proposed. Tight bounds account for the contribution of sidelobes in the likelihood ratio or the ambiguity function. Since the distances between the main lobe and the sidelobes in the likelihood function may depend on the unknown parameter, a single, parameter-independent test-point may not be enough to provide a tight bound. In the proposed class of bounds, the shift test-points are substituted with arbitrary transformations, such that the same test-point can be uniformly optimal for the entire parameter space. The use of single testpoint simplifies the bound and allows providing insight into the considered problem. The proposed bound is applied to the problem of direction-of-arrival estimation using a linear array. Simulations show that the proposed bound accurately predicts the threshold phenomenon of the maximum a-posteriori probability estimator, and is tighter than the Weiss-Weinstein bound.