See Kashiwara-Schapira.
Hovey suggests the following problem (page 199 of his book). Define “pre-triangulated category with sequential colimits”. Show: Every triangulated category gives rise to such a thing. The homotopy category of a model category is such a thing. Define a cohomology functor on such a thing, and prove that all cohomology functors on the homotopy category of a pointed cofibrantly generated model category are representable.
Rosicky: Generalized Brown Representability in homotopy categories (online at TAC), with an erratum here
Raventos and Muro has some project on representability for functors a priori defined only on compact objects, I believe. This could be of interest for example in the case of CTs defined on geometric objects but not a priori on all generalised spaces.
nLab page on Abstract Brown representability