3JS Raycaster uses the same method to calculate the uv coordinates of given point on mesh, as the rest of the world does.
To do this, you need the index of the face that target point resides on.
Basically, you need to convert the target point coordinate from Cartesian to Barycentric.
Then using the corresponding uv coords for three vertices, convert target point from Barycentric to UV coordinates.
Anyway, having the coords for target point is not enough, you also need the face index.