In [2], we propose a numerical method for fluid deformable surfaces governed by surface Stokes flow and Helfrich bending energy under active growth, aiming to model shape evolution of the epithelium sheets in developmental processes. Inspired by [3], we prevent self-intersections, which commonly arise under large deformations or low enclosed volume to area ratios, by incorporating the nonlocal tangent-point energy to penalize non-embedded configurations. The model derivation follows [5]. The resulting formulation is discretized using higher order surface finite elements similar to [1], with a parallelizable assembly strategy for the nonlocal terms. To tailor mesh quality to the geometric evolution, we propose a curvature-adaptive mesh redistribution strategy that improves mesh resolution in regions of high curvature. Numerical examples include the discocyte-to-stomatocyte transition, effectively traversing the minimizers of the Helfrich Energy as described in [4], and the inversion of a sphere within a spherical confinement and thereby demonstrate the robustness of the method in capturing large deformations, self-avoidance, and growth-induced morphology changes. Special attention is paid to mechanisms to induce the necessary symmetry-breaking for those transitions.