The HTML5 date input field is specific about the format it uses:
RFC 3339 full-date
Is it possible to change the legacy date picker to use yyyy-mm-dd format (and update the server side that reads that)?
I assume not, so then you can add some Javascript glue code that listens for a change event on the HTML5 input and updates a hidden date value field to match the format of the legacy date picker (converts yyyy-mm-dd to dd-MMM-yyyy).