When replacing an edge with a graph gadget preserves graph isomorphism?
Just a summary of my (very informal / wrong?) comments/ideas above:

Every connected gadget $GA$ with two or more nodes, that doesn't introduce an asymmetry between $u$ and $v$ should be fine. In other words suppose $maxdeg(GA)=n$, and that $X$ is a star with a central node $x$ and $n+1$ leaves, $Y$ is a star with a central node $y$ and $n+2$ leaves.

The $GA$ preserves isomorphism if and only if the two graphs $X \cup GA \cup Y$, in which $x$ is replaced with $u$ and $y$ is replaced with $v$, is isomorphic to the graph $Y \cup GA \cup X$ in which $y$ is replaced with $u$ and $x$ is replaced with $v$.

This construction can also be used to check if a small fixed $GA$ preserves isomorphism.

Furthermore, the problem of deciding if $(GA,u,v)$ preserves isomorphism is GI-complete: given two graphs $G_1,G_2$; build the following gadget $GA$: add 4 new nodes $u,x,y,v$ and edges $(u,x),(x,y),(y,v)$, then add an edge from $x$ to all the nodes of $G_1$ and an edge from $y$ to all the nodes of $G_2$. $G_1,G_2$ are isomorphic if and only if $GA$ preserves isomorphism.