If you want to keep the month heading and day under as before but just use to locale month, then you could do this.
Setup includes file to be month. No year. No day.
_includes/locale-date.html
{% assign m = page.date | date: “%-m” %}
{% case m %}
{% when ‘1’ %}Januar
{% when ‘2’ %}Februar
{% when ‘3’ %}März
{% when ‘4’ %}April
{% when ‘5’ %}Mai
{% when ‘6’ %}Juni
{% when ‘7’ %}Juli
{% when ‘8’ %}August
{% when ‘9’ %}September
{% when ‘10’ %}Oktober
{% when ‘11’ %}November
{% when ‘12’ %}Dezember
{% endcase %}
But it is trickier to get that to fit into the snippet you sent because of use of post.next and having to pass a date to the includes.