The Variational Quantum Eigensolver (VQE) is a leading algorithm for finding ground states of quantum Hamiltonians on noisy quantum devices, but requires deep circuits that are challenging for current hardware. Measurement-Based VQE (MB-VQE) offers an alternative approach using single-qubit measurements on entangled resource states. However, measurement-based quantum computing requires deterministic computation, which is guaranteed only when the measurement pattern satisfies the generalized flow (gflow) condition. This work analyzes existing MB-VQE methods and identifies gflow violations that lead to non-deterministic computation. We focus on stabilizer Hamiltonians—quantum systems whose ground states are stabilizer states—and investigate how MB-VQE performs when small perturbations are added, making the perturbed ground state unknown and requiring variational approximation methods.