Is your cousin Chloe your age? i am struggling to understand why your fmaily think it is such a problem? Why can't she be her namesake? What does it matter if there are two Chloe's 20 years apart in a family. (or 20days for that matter. ) If you really love it, i'd just use it. Why not ask your cousin?
My cousin is 13 and I don't understand it any more than you do to be honest Junglemum, but it kicked off. The thing is, its not like it would even be naming our bub after her - we just love the name Chloe! I remember when my aunt had my cousin - I was devastated she called her Chloe because even way back then I thought of it as 'my' name lol.