What's the seed mat made of? I've seen ones made of plastic--if you can remove that, please do, It's going to cause any number of issues with edging, spot maintenance, if you wish to aerate, and so in in the future.

Ones made of natural fiber like wide-spaced burlap can be left in place. While slow to rot, they do rot.

In the Ohio valley, the time to seed is...a few days ago, actually, so don't dally. :-) Optimal seeding times is when summer starts to fade, around August 15th, just as evenings begin to cool, to give the seed a week or two in warm weather to go through germination, then a few weeks of late summer weather to begin growth, then all of fall to settle in.

If you can get weedless straw, great, but even so, it tends to be rather weedy stuff. If not, peat moss (great stuff) or compost (heavy, but also great) is wonderful.

If neither are available, you can skip topdressing your seed as it's not strictly necessary and doesn't really get you out of the thrice-daily spritzing anyway.