I wash my coats all the time, not wool that are dry clean only but the ones that are just fabric.
I would probably start the water, add the detergent and some bleach, let it fill, then agitate a few times to mix it up, then add the coat. This way there won't be bleach spots on the fabric. I'd let it wash the entire cycle too, so that the bleach has plenty of time to get to the stains.
When it was done you could choose to wash it again to get all the bleach out but I'd put it in the dryer on low heat with plenty of dryer sheets so that static wouldn't be a problem afterwards.
If the stains don't come out then you have a clean coat that can be worn out to do stuff in the yard in winter or to take the kids to the park where getting it dirty won't matter. I have one old coat that I leave in the back of the van year round, in case I'm out and break down or something. I could use it as cover or as a coat to stay warm if needed.