Three-dimensional spin textures are stabilized by microscopic interactions within their host materials. X-ray vector tomography can image these nanoscale structures, but weak magnetic contrast and a limited range of viewing angles often leave parts of the magnetization poorly determined. Here we show that a reconstruction constrained by the micromagnetic energy recovers the three-dimensional magnetization from such data and gains sensitivity to the underlying material interactions. Applied to the chiral magnet Co₈Zn₉Mn₃, the method resolves the three-dimensional helical texture even from a small fraction of the dual-axis projections typically required for a successful tomographic reconstruction, or from a single tilt axis alone. By testing which ratio of exchange to Dzyaloshinskii-Moriya interaction in the reconstruction best reproduces the measured projections, we obtain a value for the sample that differs by about 10 % from the literature value derived from the helical winding period. Taken together, our findings suggest a way to reduce the time needed for vector tomography and to extend it to determining fundamental magnetic interactions in a material.