you and your family and pets will have one less threat of damage with that rotten tree down. Once you determined it was not safe, it is time to be removed. Nothing to be gained by waiting. Take it down, worry about wood later.
I can appreciate waiting a week or 2 for the ground to firm up, but as previously suggested, drop it & clear the field then go back to clean it up.
From the third picture it doesn't look too high from the ground to where the main trunk wyes. It would be pretty easy to drop both halves of the crown then take the hollow stump down after the ground hardens.
