We study the effect of local unitary noise on the entanglement evolution of a two-qubit system subject to local monitoring and interqubit coupling by constructing a suitable stochastic path integral based on the Chantasri-Dressel-Jordan formalism. From this, we identify the optimal entanglement dynamics and deploy a diagrammatic method to find a closed-form approximation for the evolution of the average squared concurrence. We find that the optimal trajectory and diagrammatic expansion capture the oscillations of entanglement at short times. A separate numerical analysis of the long-time steady-state entanglement reveals a nonmonotonic relationship between the concurrence and noise strength.