Extreme Moss!

Moss (and mushrooms) doesn't like sweet soil (pH > 7). I've had good luck applying real lime (not the pelletized limestone available big box stores). It is really dusty but works quickly.

I've heard there is a new product call pHPlus but I haven't been able to find it. Supposedly one gallon of it will do 5000 sqft and costs only $30, only twice what lime would cost. It is applied with a hose end sprayer.

Moss likes shade, water, and acid, so drainage, lime, and sunlight should get rid of it. Although I think you'd be better off figuring out how to get rid of the remaing 20% grass.

Moss likes...

Moist conditions Shade or part shade Clay soil Acidic soil

Sometimes moss is not so bad. Part of my backyard has moss growing under pine trees. Grass won't grow well there. I'd rather have the moss than bare ground. I could change that though, by removing the (large) pine trees, liming the soil, adding compost, and seeding, but I appreciate the cool shade from the pines during those 100-degree summer days. If you use some kind of commercial moss control, plan on using it on a regular basis unless you change the conditions.