I think you have a viable plan Amanda. If possible, tilling that ground first would give you a deeper seedbed for the grass roots.
Then cover with a couple of layers of cardboard, compost or composted manure and straw.
Water it well.You want the soil under the barrier to be moist as well to encourage those future grass roots to grow deep. Grass roots can grow a couple of feet into the soil if maintained correctly. Let it cook down and allow the cardboard to begin to break down over the winter and plant your seed in the spring.
This is just a basic variation of "lasagna" gardening. I've made several perennial beds this way and it works great. Mine were made over sod, but if your cardboard layer is thick and you overlap well, so there are no gaps between pieces it will smother anything underneath.
I've used horse manure as one of the layers and didn't have any horrendous weed problem as a lot of people will warn you of.
The caveat to that is not to use horse manure from horses fed bermuda hay.
If you're impatient you can sow your seed in as little as a few weeks.
You'll get a few weed seeds no matter what method you use. Weeds happen. Just stay on top of them until your grass fills in and can begin to crowd them out.
Good luck!
ETA: Make your compost combo layer nice and thick, at least 8", don't skimp because that's what your grass will be using until it gets its roots past the cardboard. This will not be a golf course lawn! But for a quick "country " lawn it's very workable.