Like several previous posters, it doesn't sound like a badger to me. I have heard all sorts of noises from them - including 'chattering'when they are with other members of their clan - but not nothing like loud breathing.
What is on the ground outside the window? If it is grass or soil, I would expect a regularly-visiting badger to be leaving some visible clues out there, either scratched-up grass/moss (or even small holes that people often think have been made by squirrels) where they have foraged for worms and grubs, or shallow indentations hollowed out in the grass or soil with their poo in them.
I don't know enough about hedgehogs to comment on the noises they can make, but if they are out there, they too are likely to be leaving some very distinctive poo lying around which is completely different from that of a badger.
Personally, I would be out there looking for those clues - I do enjoy a good garden mystery every now and then. ☺