We continue our investigations of the magnetorotational collapse of stellar cores by discussing simulations performed with a modified Newtonian gravitational potential that mimics general relativistic effects. The approximate TOV gravitational potential used in our simulations captures several basic features of fully relativistic simulations quite well. In particular, it is able to correctly reproduce the behavior of models that show a qualitative change both of the dynamics and the gravitational wave signal when switching from Newtonian to fully relativistic simulations. For models where the dynamics and gravitational wave signals are already captured qualitatively correctly by a Newtonian potential, the results of the Newtonian and the approximate TOV models differ quantitatively. The collapse proceeds to higher densities with the approximate TOV potential, allowing for a more efficient amplification of the magnetic field by differential rotation. The strength of the saturation fields (∼10^15 G at the surface of the inner core) is a factor of two to three higher than in Newtonian gravity. Due to the more efficient field amplification, the influence of magnetic fields is considerably more pronounced than in the Newtonian case for some of the models. As in the Newtonian case, sufficiently strong magnetic fields slow down the core’s rotation and trigger a secular contraction phase to higher densities. More clearly than in Newtonian models, the collapsed cores of these models exhibit two different kinds of shock generation. Due to magnetic braking, a first shock wave created during the initial centrifugal bounce at subnuclear densities does not suffice for ejecting any mass, and the temporarily stabilized core continues to collapse to supranuclear densities. Another stronger shock wave is generated during the second bounce as the core exceeds nuclear matter density. The gravitational wave signal of these models does not fit into the standard classification. Therefore, in the first paper of this series we introduced a new type of gravitational wave signal, which we call type IV or “magnetic type”. This signal type is more frequent for the approximate relativistic potential than for the Newtonian one. Most of our weak-field models are marginally detectable with the current LIGO interferometer for a source located at a distance of 10 kpc. Strongly magnetized models emit a substantial fraction of their GW power at very low frequencies. A flat spectrum between 10 Hz and <∼100 kHz denotes the generation of a jet-like hydromagnetic outflow.