I would start with a rake and scrape it off. Then apply the Moss Out.

Well you kind of jumped the gun, I can understand getting rid of the moss, but to do so 5 months before you plan on seeding wasn't the best idea. What's done is what's done I suppose. The problem with bare spots is they tend to fill in with weeds, and undesirable grass like poa annua and poa trivialis. Here's what I would do if I was you. Apply Dimension preemergent now. It should last 4 months, preventing any weed from maturing. I would apply a layer of mulch to bare spots to prevent erosion. I would treat the lawn like you normally would, getting it ready for your late summer overseed (a soil test now would be helpful). Kill any weeds that pop up with spot applications of the appropriate herbicides. In August, kill everything with Round-up, and any remaining moss with Moss out. Once it's all dead, then scalp it all down with a mower, and seed. Or you can cut it short, Round up, and then seed the next day, it's up to you. I remember you from last year, and you have shade issues, please remind us of how much sun you get on average during the growing season, and what your square footage is. I just renovated my shady backyard last year with a grass that might be a perfect fit for you.

That's a pretty common occurance. Peat moss is hydrophobic, but not permanently. Usually when you topdress with peat moss it helps if you do light waterings in five minute cycles, a few times for the first day. So if you have 4 watering zones, do five minutes at each zone 3 or 4 times. After that you can resume your normal watering cycle.

I think you'll find the area where you see moss is more damp with moisture than outside of the area. Probably a depression is there allowing wetness to remain longer and maybe too isn't affected by winds that can dry out areas well. I think sometimes they suggest the lawn has been left to its own devices....you have maybe ignored feeding it fertilizer. Give the area a good stiff raking, moving the rake from the outside of the area to the center so it doesn't spread the spores. Then dispose of....but not in the compost. Sunlight too...but moss will grow in sun...but prefers shade. So if you can make the area more open to sunlight that should help. The compost will help too, feed the soil, make the grass grow and push out weeds. An overseeding wouldn't harm either.