Top dressing with peat is never a good idea. First, it is overly acidic and no lawn needs or requires that much acidity. Second, it is very lightweight and does wash away easily with rain. And finally, if allowed to dry out, it becomes hydrophobic and is almost impossble to rewet.

You do want to topdress to protect the seed and most importantly, maintain enough moisture for germination. But just a thinnish layer of soil or screened compost should work well.

Yeah, I know P is light - too light and would dry out easily bearing how thin the application was. Figured as a cover-topdress it would be easy for the blade to get through the topdress (if both would just stay around long enough to not get washed out)

Farmers have noticed that when livestock steps on seed, the seed sprouts better in the footprint. All you need to do is make good contact between the seed and soil so the moist soil can keep the seed moist longer.