Does the existence of an injective cogenerator "help" in finding generators of an abelian category?

A few thoughts a bit too long for a comment. I should preface this by saying that I'm not that familiar with the sort of generator conditions you're using, so I can only really talk about analogous cases using stronger generating conditions. I'm really hoping that someone who actually knows these things might respond to this.

1. As a cautionary analogy, consider the fact that the opposite of a locally presentable category is never locally presentable unless the category is a poset. A locally presentable category is the a cocomplete category $\mathcal{C}$ with the nicest possible kind of generator: a small dense generator $\mathcal{G}$. "Dense" means that the restricted Yoneda embedding from $\mathcal{C}$ into presheaves on $\mathcal{G}$ is fully faithful (unless you assume Vopenka's principle, you have to additionally assert that the objects of $\mathcal{G}$ are presentable, i.e. their hom-functors preserve sufficiently-filtered colimits. In fact, there's a weaker formulation of the generating property which turns out to be equivalent: the generator should be strong, i.e. the hom-functors are jointly faithful and jointly conservative, and consist of presentable objects). So the fact that the opposite of a locally presentable category is never locally presentable puts restrictions on how "nice" a generator and a cogenerator can simultaneously be. I'm not sure whether such restrictions extend to weaker notions of generator.

   My sense is that most triangulated categories, like the homotopy category, don't have many (co)limits, and so are not locally presentable -- but moreover, like the homotopy category I wouldn't expect a triangulated category to even admit a faithful functor to $\mathsf{Set}$, so it's not even accessible. For this reason, I suppose, the study of triangulated categories uses weaker notions of generator than you see elsewhere in category theory. So maybe this whole point is moot. But I take it you're working with something abelian, more like the heart of $t$-structure, so my intuition is not clear on whether such a category is likely to be locally presentable. But somebody out there surely knows whether hearts of $t$-structures tend to be locally presentable! And somebody surely knows whether there's a tension between the generating and cogenerating properties you're using.

2. As a positive analogy, Giraud's theorem is a case where the dual implication of the one you want holds: the existence of a generator in a category with certain properties implies the existence of a cogenerator (the subobject classifier). I think the construction of the subobject classifier uses the full force of Giraud's theorem: you have to know that your category embeds into the category of presheaves on the generator.

   But I suppose this analogy is probably no more enlightening than the usual construction of injectives in abelian categories of sheaves.

3. As another positive analogy, the Freyd-Mitchell embedding theorem is another place with a weird interplay between generators and cogenerators. To embed a small abelian category $\mathcal{A}$ exactly into a category of modules (which has a nice generator), you first embed it exactly (via Yoneda) into the category of pro-objects $\mathrm{Pro}(\mathcal{A})$, i.e. the opposite of the category of left-exact functors $\mathcal{A} \to \mathsf{Ab}$ (a category with a reasonable cogenerator given by $\mathcal{A}$ itself), and do some further work I'm not familiar with to embed nice subcategories of $\mathrm{Pro}(\mathcal{A})$ into categories with nice generators.