Engineering design optimization problems often have two competing objectives as well as uncertainty. For these problems, quite often there is interest in obtaining feasibly robust optimum solutions. Feasibly robust here refers to solutions that are feasible under all uncertain conditions. In general, obtaining bi-objective feasibly robust solutions can be computationally expensive, even more so when the functions to evaluate are themselves computationally intensive. Although surrogates have been utilized to decrease the computational costs of such problems, there is limited usage of Bayesian frameworks on problems of multi-objective optimization under interval uncertainty. This article seeks to formulate an approach for the solution of these problems via the expected improvement Bayesian acquisition function. In this paper, a method is developed for iteratively relaxing the solutions to facilitate convergence to a set of non-dominated, robust optimal solutions. Additionally, a variation of the bi-objective expected improvement criterion is proposed to encourage variety and density of the robust bi-objective non-dominated solutions. Several examples are tested and compared against other bi-objective robust optimization approaches with surrogate utilization. It is shown that the proposed method performs well at finding robustly optimized feasible solutions with limited function evaluations.