From the picture of the toy model it should be clear that there is axial symmetry. We already mentioned this when this toy model was first introduced. The distance between the electron-charges and the nuclear dumb-bell charges does not depend on the orientation of the dumb-bell in the xy-plane. So there’s no reason that the electric field gradient be dependent on this.
I think I have to specify that the nuclear dumb-bell does not feel a different electric gradient field depending on its precise position for a specific angle theta as the distance between the nuclear dumb-bell charges and the electron-charges does not change.