@sinnbeck this worked for me:
console.log(flatpickr.formatDate(new Date(2022, 06, 03), 'F j, Y', flatpickr.l10ns.da))
passing the locale as third param on the formatDate function, and I had to add the import
<script src="https://npmcdn.com/flatpickr/dist/l10n/da.js"></script>
not sure why the global one does not work, at least not in JSFiddle.