The measurement of full-field displacement data based on image correlation techniques like global digital volume correlation (GDVC) allows a more detailed analysis of specimens in material sciences, as additional data is obtained in comparison with more traditional approaches. The robustness and accuracy of the computed full-field displacements can be improved by applying mechanical regularization, which aims for compliance of the computed displacements with mechanical equilibrium laws. This aim is achieved by adding an additional term considering the mechanical equilibrium of the external and internal residual forces to the objective function of image correlation. To efficiently allow for heterogeneous materials, the finite cell method is considered here. However, for the computation of the internal residual forces, the material parameters have to be known, which creates a strong constraint for the application of the regularization. Therefore, we propose an enhanced mechanically-regularized DVC, that builds on the iterative application of the GDVC with fixed material parameters to determine a three-dimensional displacement field and a subsequent application of the equilibrium gap method (EGM) to compute updated material parameters for the now fixed displacements. Repeating these two steps, a convergence of both the displacement field as well as the computed material parameters can be obtained. At the same time, the mechanical regularization of the GDVC becomes more robust due to the update of the material parameters. The application of the EGM does not require the solution of forward finite element problems and is thus a highly efficient way of simultaneously identifying material parameters. We validate the proposed method using a synthetic dataset based on a single CT image of an unloaded state. The associated deformed state is computed by artificially deforming the CT image using the deformation from a forward finite element computation, allowing for a qualitative analysis of the proposed method, since, in contrast to an experimental setting, the correct solution is known from the forward simulation.