Kids can deal perfectly well with different people in their lives calling them by different names. And this goes double because your child is only 10 months old -- she won't have any difficulty with this.
It certainly is not "weird" to call a child by a short form of a middle name. Back in the 19th century a rather high percentage of Americans were called in everyday life by the name which was in the middle position on their birth certificates. My paternal grandfather was officially named "
Clarence Kent" but always called
Kent. My maternal grandfather was officially "
Grover Cleveland" but always called "
Cleve." My mother's oldest sister was "
Elizabeth Lucille" and always called
Cille or
Lucille. My first cousin closest to my age is officially
Mary Eugenia and has always been called
Gena. There are literally millions of people in English speaking countries who've lived since middle names first became common in the 19th century who have gone by their middle name in everyday life.
Now, if your daughter and her cousin are going to have frequent interaction, it might on occasion be confusing to have them called
Jewel and
Jules. It will probably be more confusing for other people than them, though.