The relation $R$ can be empty; _i.e._, if $\kappa$ is the least ordinal such that $L_\kappa(\mathbb{R})$ is admissible, then the structure $(L_\kappa; \in, \emptyset)$ is a companion of the pointclass $\mathrm{IND}$ of inductive sets.  We can prove a somewhat more general statement.

Assume that $\kappa$ is an ordinal such that

- $J_\kappa(\mathbb{R})$ is admissible (in which case $J_\kappa(\mathbb{R}) = L_\kappa(\mathbb{R})$, but there may be points later in the argument where it is necessary to use the Jensen hierarchy) and

- $\kappa$ begins a $\Sigma_1$-gap in $L(\mathbb{R})$, meaning that $J_\alpha(\mathbb{R})$ is not a $\Sigma_1(\mathbb{R} \cup \{\mathbb{R}\})$-elementary substructure of $J_\kappa(\mathbb{R})$ for any ordinal $\alpha <\kappa$. 

Then the structure $(L_\kappa; \in, \emptyset)$ is a companion for the pointclass $\mathbf{\Sigma}_1^{J_\kappa(\mathbb{R})}$. (So letting $J_\kappa(\mathbb{R})$ be the least admissible level of $L(\mathbb{R})$ we obtain the desired result for the pointclass $\mathrm{IND}$.)

The criterion for "companionship" that was not clear to me when posting the question was projectability:  the existence of a $\mathbf{\Delta}_1^{J_\kappa(\mathbb{R})}$ partial surjection $\mathbb{R} \dashrightarrow J_\kappa(\mathbb{R})$.  The existence of a $\mathbf{\Sigma}_1^{J_\kappa(\mathbb{R})}$ partial surjection $\mathbb{R} \dashrightarrow J_\kappa(\mathbb{R})$ is well-known.  It turns out that if we use a bit of care when defining the partial surjection then it will be $\mathbf{\Delta}_1^{J_\kappa(\mathbb{R})}$ (although its domain can only be $\mathbf{\Sigma}_1^{J_\kappa(\mathbb{R})}$.)

Letting $(\varphi_n : n<\omega)$ be an effective enumeration of $\Sigma_1$ formulas, we define a partial surjection $F:\omega \times \mathbb{R}\times \mathbb{R} \dashrightarrow J_\kappa(\mathbb{R})$ by setting $F(n,x,y) = S$ if there is an ordinal $\alpha < \kappa$ such that

- $\alpha+1$ begins a gap as witnessed by $\varphi_n$ and $x$, meaning that $\varphi_n[x,\mathbb{R}]$ holds in $J_{\alpha+1}(\mathbb{R})$ but not in any earlier level of the Jensen hierarchy.

- $S \in J_{\alpha+1}(\mathbb{R})$.

- For all $\beta < \alpha$, if $\beta+1$ begins a gap then $S \notin J_{\beta+1}(\mathbb{R})$

- $f_{\alpha+1}(y) = S$ where $f_{\alpha+1}$ denotes the canonical partial surjection $\mathbb{R} \dashrightarrow J_{\alpha+1}(\mathbb{R})$ that exists because $J_{\alpha+1}(\mathbb{R})$ must project to $\mathbb{R}$ in order for $\alpha+1$ to begin a gap (the level of definability of $f_{\alpha+1}$ over $J_{\alpha+1}(\mathbb{R})$ is not important.)

It is straightforward to show that this $F$ is indeed a partial surjection and is $\mathbf{\Sigma}_1^{J_\kappa(\mathbb{R})}$.  But in fact it is $\mathbf{\Delta}_1^{J_\kappa(\mathbb{R})}$, because to check that $F(n,x,y) \ne S$ we only have to find some $\Sigma_1$-gap that begins past the place in the Jensen hierarchy where $S$ appears and verify some local statement there.
Identifying $\omega \times \mathbb{R}\times \mathbb{R}$ with $\mathbb{R}$, we have the desired $\mathbf{\Delta}_1^{J_\kappa(\mathbb{R})}$ partial surjection $\mathbb{R} \dashrightarrow J_\kappa(\mathbb{R})$ witnessing projectability.