Total Variation (TV) regularized reconstruction is one of the most relevant methods to improve the quality of limited-angle tomographic reconstructions. Nevertheless, the accuracy of computed tomography (CT) reconstructions with a limited number of measurements can be further improved by selecting the most informative acquisition angles. This optimal experimental design (OED) task can be formulated as a bi-level optimization problem, with selecting optimal angle combinations (experimental design parameter) on the upper-level and tomographic reconstruction on the lower-level. However, integrating TV regularized reconstruction into the bi-level optimization approach is non-trivial because of the large number of iterations required for the algorithm convergence, which impedes naive computation of gradients of the upper-level objective with respect to the experimental design parameter. In this work, we address this problem by employing implicit differentiation approach to calculate the upper-level objective gradient. Moreover, we utilize inexact methods to dynamically adjust the accuracy of the lower-level solver, refining the gradient calculation as needed. We demonstrate that this approach makes OED with TV regularized reconstruction applicable to realistic 3D data. Our numerical results demonstrate that the angles selected by our bi-level optimization framework significantly outperform the standard equidistant angular selection. The proposed approach is therefore effective in minimizing experimental time and radiation dose requirements for CT reconstruction of objects benefiting from TV regularization, and can be readily extended to other types of computationally demanding iterative reconstruction algorithms.