If this map just changes $2$ and $3$ is it a graph automorphism? It's not clear to me whether or not there should be an edge between $1$ and $2$ in the map's image because it isn't a geometrical symmetry, as in a horizontal flip of a rectangle, without repositioning or moving the vertices in the ambient space so the angles & lengths are unchanged.
It isn't in the graph automorphisms listed in this tutorial: https://www.youtube.com/watch?v=X4_4Bqj6EdA&t=368s and more specifically the definition "$uv$ is an edge in $G$ iff the permutation $p(u)p(v)$ is an edge in $G$" looks like it is saying $1,2$ is not an edge in G because $p(1)p(2) = 1,3$ is not an edge in $G$... the edge $1,2$ obviously must exist because it is in the original graph $G$ so why should the definition mean $1,2$ is not an edge?
One very basic step in an automorphism is that only vertices of the same degree can be mapped to each other. So there will be no automorphism that maps $2$ to $3$ or vice versa.
The next basic step is that adjacent vertices should be mapped to adjacent vertices, etc. - this proposed swap will again fail this test as $2$ would no longer be adjacent to $1$. What you shouldn't do - as you appear to propose - is change the structure (the connectivity) at the same time as you change the labels.