double barrel names with elizabeth?

Hi! My husband and I are expecting a girl in May but so far we're having a hard time with her name. Might be a bit of an unpopular opinion but we really like double barrel names. We're thinking we'd like to give the baby one and use Elizabeth as part of it (as first half or last, not attached to either). We just don't think anything sound goods with it! A lot of the ones we tried out run together with the "uh" sound or just end up sounding awkward together to say out loud. One of the top contenders is Elizabeth Kate but we're not totally sold on it. We like a lot of sweet, classic names like Emma, Alice, and Katherine. The middle name will probably be Hana and the last name is very common, two syllables. I thought I'd see if anyone has any ideas or advice for us.