You are essentially asking for a local cross section for the fiber bundle $(G,\pi: G \to G/H)$. According to this article, it is known that such a section exists if the group $G$ is Lie, or more generally, LCSC and with finite covering dimension. I'm not an expert on the subject, but it seems to me that there exist metrizable LCSC abelian groups with infinite covering dimension, so this doesn't seem to give a complete answer to your question. Perhaps someone who knows more on the subject can illuminate this point, or provide a more recent reference.

You are essentially asking for a local cross section for the fiber bundle $(G,\pi: G \to G/H)$. According to this article¹, it is known that such a section exists if the group $G$ is Lie, or more generally, LCSC and with finite covering dimension. I'm not an expert on the subject, but it seems to me that there exist metrizable LCSC abelian groups with infinite covering dimension, so this doesn't seem to give a complete answer to your question. Perhaps someone who knows more on the subject can illuminate this point, or provide a more recent reference.

¹On the Universal Space for Group Actions with Compact Isotropyby Wolfgang L Lück and David Meintrup