I find this interesting, and so far haven't found anything in an internet search.
The Moon’s rotational period is the same as its orbital period: the points on its surface are (approximately) always in the same relationship to Earth. Because of this it is convenient to switch to a frame that rotates with the Earth-Moon system.
Then we're in luck, because now the problem has already been exhaustively explored for us!
Euler found the collinear stationary points L₁ through L₃ and Joseph-Louis Lagrange added the triangular stationary point L₄ and L₅ thereby completing the picture and showing mathematically that these are the only five stationary points in a CR3BP or circular restricted three-body problem. This was briefly mentioned elsewhere as well.
So in a word, no.
For a tidally locked body there are not going to be any points stationary to the surface other than Lagrange points.
If the Moon was spinning (much) faster (i.e. a long time ago), then a stable synchronous orbit may have been possible. But I’m not sure it was ever spinning that fast.