I d go with the woods option. Are there children insisting on a proper burial? Can you drop it off at a vets for disposal?
Perhaps use a tablespoon rather than a spade in the woods. After all this rain the ground won't take much digging? Do you have access to a river or stream , give it a water burial, float it downstream with a garland of flowers?
I live in a flat, our hamster died, my DD wanted to bury it, so i got a big flower pot and a flower we buried the hamster in the pot with the flower on top, when she went to bed i dug up binned the poor hamster, she was happy and im sure the hamster didn't mind, we still have the flower that she thinks has grown over her dead hammy,