If you are looking for results in the literature you need to be sure you are using a definition that others use. Do you have a reference for your definition of $k$-iso-regular? The definition I am familiar with is that for any set of up to $k$ vertices the number of common neighbors depends only on the isomorphism type of the induced graph on those vertices so:

Every vertex has the same number of neighbors (aka regular) AND the number of common neighbors of two vertices $u,v$ depends only if $uv$ is or is not an edge (aka strongly regular) AND the number of common neighbors of $u,v,w$ depends on the isomorphism type ($4$ cases) of the induced graph on $u,v,w$ etc up to $k$ vertex sets.

If you are looking for results in the literature you need to be sure you are using a definition that others use. Do you have a reference for your definition of $k$-iso-regular? The definition I am familiar with is that for any set of up to $k$ vertices the number of common neighbors depends only on the isomorphism type of the induced graph on those vertices so:

Every vertex has the same number of neighbors (aka regular) AND the number of common neighbors of two vertices $u,v$ depends only if $uv$ is or is not an edge (aka strongly regular) AND the number of common neighbors of $u,v,w$ depends on the isomorphism type ($4$ cases) of the induced graph on $u,v,w$ etc up to $k$ vertex sets.

also For graphs of bounded degree ,GI is decidable in polynomial time.