Having the snout full means to be fed up with it, to be sick to death of it.

And the German Greens are filling up everybody’s snouts these days.
Germans Fed Up With Greens as Founding Member Resigns – The aggressive policies of the German Greens have alienated not only many voters in Germany but also their own supporters, including one of the founders of the party who has accused them of pursuing a warmongering foreign policy…
Annalena Baerbock, the young, inexperienced minister of foreign affairs, has promised to introduce a ‘feminist foreign policy,’ but she seems helpless in the face of the momentous challenges posed by the Russian-Ukraine war. The Greens have converted from a ‘pacifist’ party into the most belligerent party of all.