I still have the badger in my garden although it isn't digging up the grass so much now.
My problem now is that I have a new neighbour just moved in and he fills the hole up with stones and goodness knows what everytime the badger comes in and each time the badger digs them out and makes an even bigger hole.
I would prefer to just leave the holes and let it come and go as it pleases in the hope that it will eventually move on.
You are so lucky, your lawn may not think so, or your neighbours, i wou ld let them be, mating season is now over late december to end feb is birth seasons, , you most probably know this but when the badger mates it will hold the sperm until a time were they know that the babies should strive, and then release the sperm to the egg, to give the cub the best chance of life.