Gf refuses to treat her chronic cough on religious grounds. So what do I do?

Personal choice. I'm not sure what 'religious grounds' would stop her from taking good care of her temple but, maybe she has her own reasons. In the end, you can't force her to get evaluated but, you can tell her how her coughing affects you and the relationship. If she chooses not to address it, then you are the one who needs to make a choice- to be with her and not talk about the coughing or to move on...