If I understand your question, then no. While the Earth's J2 (oblateness) produces enough torque to rotate a Sun-synchronous orbit once a year, it does not produce enough torque to rotate a "Moon-synchronous orbit" once a month. So there is no such orbit.

I am not clear on what the utility of such an orbit would be, even if it did exist. If you're going to point antennas at the Moon from the near vicinity of Earth, you can have many more antennas and much larger antennas on the surface of the Earth than what you would be able to put into Earth orbit for an equal amount of money. You would only need to divide your total aperture by three to account for the downside of not always being able to see the Moon.

If I understand your edited question, then no. While the Earth's J2 (oblateness) produces enough torque to rotate a Sun-synchronous orbit once a year, it does not produce enough torque to rotate a "Moon-synchronous orbit" once a month. So there is no such orbit.

I am not clear on what the utility of such an orbit would be, even if it did exist. If you're going to point antennas at the Moon from the near vicinity of Earth, you can have many more antennas and much larger antennas on the surface of the Earth than what you would be able to put into Earth orbit for an equal amount of money. You would only need to divide your total aperture by three to account for the downside of not always being able to see the Moon.