Estimating unknown parameters subject to prior constraints is important in statistical inference, particularly in fields such as reliability analysis, survival studies, and engineering, where prior structural information about the parameters is often available. Incorporating such prior information makes the analysis more realistic and usually yields better estimates than methods that ignore such information. In this article, we consider the problem of estimating the positive power of the scale parameter of a two-shifted exponential population under a prior ordering constraint on scale parameters. We derive sufficient conditions under which equivariant estimators are shown to dominate others under scale-invariant strictly convex loss functions. In addition, we derived various estimators that dominate the best affine equivariant estimators (BAEE). Moreover, we derive a smooth estimator which dominates the BAEE using an integrated approach, and we further show that it is a generalized Bayes estimator under a non-informative prior. We also provide an improved estimator based on the Pitman closeness criterion. An extensive simulation study has been done for computational purposes. Finally, we provided real examples to implement the results.