Moon, rocket, Earth and Sun aligned in this order (the most favorable case).
2364.3 m/s
Moon, rocket and Earth aligned in this order (without considering the effect of the Sun).
2365.2 m/s
Moon and rocket (result less approximate).
2373.3 m/s
Data used:
ML = 7.34E22 kg
RL = 1739E3 m
MT = 5.98E24 kg
MS = 1.99E30 kg
DTL = 3564E5 m
DLS = 146E9 m
G = 6.67259E-11