Just see your other post so badger Q being discussed, though if it were me I would encourage them to move elswhere, (after losing ducklings and lots of eggs, even goose eggs being found down near a sett) they don't like Jeyes fluid sprinkled round.
My geese have quite a range, certain areas they seem to keep fairly short, but they are often in the longer grass as well, not had a problem with impaction.
If you are rotating the other animals I should think the geese will take advantage of the shorter grass left behind. You could occasionally mow small areas for them?
An old saying - '3 geese eat as much as a cow',
Geese and orchards are traditional, I think it sounds good.
Apart from the badgers
How close to the external fence is the sett? could you fence a way straight to an 'exit'? Or would a neighbour be a bit annoyed?
Badgers don't like a low electric fence either, but grass would have to be kept short. Rutland fencing website has some info.