Approximating Independent Dominating set on bipartite graphs
After a bit more searching, it appears that what I'm looking for is unlikely to exist.
In [1], it is proven that approximating the minimum maximal independence number (which is equivalent to the minimum size of an independent dominating set) within a factor of $O(n^{1-\epsilon})$ is $\mathrm{NP}$-hard for any $\epsilon &gt; 0$. This remains true even when restricted to bipartite graph (this is the part I missed when I first came across this paper).
[1] Halldórsson, Magnús M., Approximating the minimum maximal independence number, Inf. Process. Lett. 46, No. 4, 169-172 (1993). ZBL0778.68041.