I'm editing this to reflect the discussion. 

You might think that there cannot be an automorphism of ${\mathbb R}P^3$ that acts non-trivially on the degree one cohomology. But the bijection between the spin structures and degree one cohomology is not canonical, it's a torsor. So it may be possible for an automorphism to swap the two spin structures.

To fix conventions, a spin structure is a structure on a manifold with a given orientation. The space ${\mathbb R}P^3$ is orientable because $3$ is odd, and there is an orientation reversing diffeomorphism which gives a bijection between the two spin structures for one and the two spin structures for the other. So we fix an orientation.

Now in Section IV.B.(i) of Dabrowski and Trautman, "Spinor structures on spheres and projective spaces", in the second to last paragraph they give a topological way to distinguish the two spin structures, so there cannot be an orientation preserving diffeomorphism swapping them. 

They then go on to make the claim that the two spin structures are related by an orientation reversing isometry. This I do not understand, and it misled me to give my original answer.