That would be a tough call. You have a lot of area to work with, but you need to get from Point A (mud hole) to Point B (green) and they will interfere with that effort at every turn.
If you are looking a a mono-culture, like a turf grass, you may be looking in the wrong direction. What you may want to consider instead is a large variety of plantings. Gardens, clovers, annuals, perennials, shrubs, bushes, tall ornamental grasses if they will grow there, etc. Maybe turf grass alleys all around to direct them. You may have to cordon off one area to get plantings started and confine them to a much smaller area until they can move in. In the short run, confine them to a fraction of that and use deep bedding like old hay, etc. Start with 6" and build from there.
Basic concept is to create a lot of edge and variety. Tough upright things they cannot reach or destroy.