@MarcHoyois Ah, I see. This is similar to
Denis-Charles' answer to Charles' related MO question. Why do we need the representables to be hypercomplete? Upon reflection, it seems to me that if $i^\ast: E \to LE : i_\ast$ is any cotopological localization, then $i_\ast$ preserves effective epimorphisms, so as you say, any object $X$ of $E$ not in $LE$ yields a functor $Hom(i_\ast -, X): LE^{op} \to Top$