(The following doesn't address the precise version of Brown representability asked about in the question body, but it does address _some_ version of the title question, so I think for searchability it should be useful to mention it here.)

In Prop 2.15 of [Adams spectral sequences and Franke's algebraicity conjecture](https://arxiv.org/abs/2110.03669), Patchkoria and Pstragowski prove the following:

**Thm (Patchkoria and Pstragowski):** Let $C$ be a presentable stable $\infty$-category and let $E : C^{op} \to  Ab$ be a homological functor which takes arbitrary direct sums in C to products of abelian groups. Then, $E$ is representable in the homotopy category $hC$.