Index sets for $n$-decidable structures categorical relative to $m$-decidable presentations

We say that a structure is categorical relative to $n$-decidable presentations (or autostable relative to $n$-constructivizations) if any two $n$-decidable copies of the structure are computably isomorphic. For $n=0$, we have the classical definition of a computably categorical (autostable) structure. Downey, Kach, Lempp, Lewis, Montalbán, and Turetsky proved that there is no simple syntactic characterization of computable categoricity. More formally, they showed that the index set of computably categorical structures is $\Pi^1_1$-complete.
We study index sets of $n$-decidable structures that are categorical relative to $m$-decidable presentations, for various $m,n\in\omega$. If $m\ge n\ge0$, then the index set is again $\Pi^1_1$-complete, i.e., there is no nice description of the class of $n$-decidable structures that are categorical relative to $m$-decidable presentations. In the case $m=n-1\ge0$, the index set is $\Pi^0_4$-complete, while if $0\le m\le n-2$, the index set is $\Sigma^0_3$-complete.