As long as you don't do a lot of scrubbing or agitation, the wool should do ok. One thing to be aware of is that wool will also felt from changes in temp of the water between wash and rinse. The recommendation when washing wool is to have the water at room temperature.
If I were washing this, I would have a flat sheet ready to use to move the quilt out of the tub. Never lift that baby while it's wet withput something underneath to support it. I'd be afraid that sateen would be too weak to tske the strain and would tear. That's the other reason I am not advising using a washer of any type. Also, I would soak it in whatever solution you are using for at least a full day before I did anything else. There will probably be a lot of dirt in the water, but don't worry about that. Whatever detergent or soap you use will keep it suspended in the water. When you want to go from wash to rinse, pull the plug and let it drain for at least an hour. Then gently press it to remove the majority of the wash solution. When you fill the tub, make sure you do not let the water run directly onto the quilt, as that is enough to felt the wool where the water is hitting it.
Another thing no one has covered yet that the rear end of a sheep is a gross place! So I wouldn't worry too much about any puppy products and the filling. It will wash out just like the sheep stuff did when that fleece was fresh off the sheep. You are more likely to have problems getting it off the cover than the wool batting.
Good luck! I wish we lived closer together, I'd help you with this- I love a wool related challenge!