I voted no. I think Cora Elizabeth would be really pretty.
I don't think you should feel pressure from yourself/your child to use your MN if you dislike it. But have you considered adding a similar suffix to the names you already like? Like Coraline or Emmeline? You could still use the NNs Cora and Emma, but maybe the modern suffix would give you a pass in your mind like you are giving your daughter a family name and not just random names you like.
(Although isn't Day a family name for you? If I am remembering that correctly I think that as the MN would fulfill any 'obligations.'