I love
Millie but i really don't like
Mildred.
If you wanted to use
Millie as her name but still have a full name for her you coud use an alternative of
Millicent. You would still be honoring your grandmother by using
Millie.
I know a gorgeous little girl with the name
Millicent but she always goes by
Millie.
This way she would have a full name to fall back on when shes older.
I liked the suggestion of
Rose as a middle name:
Millicent Rose (nn
Millie).
Hope this helped.