Hi @floriz
This is an interesting issue which I would like to solve and write a tutorial for.
Here is an older thread which covered some ideas / solutions:
http://old.support.advancedcustomfields.com/discussion/4789/date-field-output-in-non-english-language
I wonder if it is posisble to use the WP __() translate function to translate the month like so:
$month = $date->format(‘M’);
echo __($month);