First, you should make clear whether LCA groups are assumed to be Hausdorff or not.
Next, whichever way you answer the previous question, the category of LCA groups will not be an abelian category. Indeed, it is a standard lemma that in an abelian category, every morphism that is both a monomorphism and an epimorphism has an inverse. If we allow LCA groups to be non-Hausdorff then the identity map $$(\mathbf{Z},\text{discrete topology}) \to (\mathbf{Z},\text{indiscrete topology})$$ is both mono and epi but has no inverse. If instead we restrict attention to Hausdorff In LCA groups, then every homomorphism with dense image is a categorical epimorphism. It follows that the identity map $$(\mathbf{R},\text{discrete topology}) \to (\mathbf{R},\text{usual metric topology})$$ is both mono and epi, but there is no inverse.
If you want to work with abelian categories, then the theory of torsion theories (see http://ncatlab.org/nlab/show/torsion+theory for example) does more or less what you ask for. However, this theory would need to be extended before it could cover the nonabelian category of LCAs.
