We report on the adiabatic splitting of a Bose-Einstein condensate of 87Rb atoms by an asymmetric double-well potential located above the edge of a perpendicularly magnetized TbGdFeCo film atom chip. By controlling the barrier height and double-well asymmetry, the sensitivity of the axial splitting process is investigated through observation of the fractional atom distribution between the left and right wells. This process constitutes a novel sensor for which we infer a single shot sensitivity to gravity fields of delta g/g ≈2×10-4. From a simple analytic model, we propose improvements to chip-based gravity detectors using this demonstrated methodology.