I suggest save time and labor. Apply the seed first, then rake it vigorously to work the seed into the soil. Add compost if you wish. Done. For small spots seed cost is not important--use plenty of seed--triple the recommended rate is not too much. Seed is cheaper than labor (for small areas).
Remember in the south they over seed Bermuda on golf courses with about 20 pounds of perennial rye per thousand sqft.
Late August should work if the temps have come down below 85. Germination is fast because the soil temp is high--about 75. September also works, as well as October and even November, (judging from my tests last year).
Soil temp averages: