(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, 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 $hC$.