The bef_datepicker plugin is enough to play with dates but it would be nice if we could also play with times.
Proposed resolution
Add an advanced filter option to "Date picker" plugin : "Allow the user to set time"
When the filter is enabled, add two numeric fields under the calendar in order to allow the user to set hours (min: 0, max: 23) and minutes (min: 0, max: 59)
Convert offset initial values to datetimes using those values $date->format('Y-m-d H:i')