I came across this when looking for a purpose built, fully customisable calendar – https://fullcalendar.io/
Im just starting to work with it now for an in-house app/web design project that needs the option of populating the calendar with a shift pattern using a JSON feed. The user will be able to request a shift swap on a selected day posting via a private custom post type.
Very impressed so far and if your looking for a fully customisable calendar, this is pretty good!