Quantitative oblique back-illumination microscopy (qOBM) is an emerging label-free optical imaging technology that enables 3D, tomographic quantitative phase imaging (QPI) with epi-illumination in thick scattering samples. In this work, we present a robust optimization of a flexible, fiber-optic-based qOBM system. Our approach enables in silico optimization of the phase signal-to-noise-ratio over a wide parameter space and obviates the need for tedious experimental optimization which could easily miss optimal conditions. Experimental validations of the simulations are also presented and sensitivity limits for the probe are assessed. The optimized probe is light-weight (∼40g) and compact (8mm in diameter) and achieves a 2µm lateral resolution, 6µm axial resolution, and a 300µm field of view, with near video-rate operation (10Hz, limited by the camera). The phase sensitivity is <20nm for a single qOBM acquisition (at 10Hz) and a lower limit of ∼3 nm via multi-frame averaging. Finally, to demonstrate the utility of the optimized probe, we image (1) thick, fixed rat brain samples from a 9L gliosarcoma tumor model and (2) freshly excised human brain tissues from neurosurgery. Acquired qOBM images using the flexible fiber-optic probe are in excellent agreement with those from a free-space qOBM system (both in-situ), as well as with gold-standard histopathology slices (after tissue processing).