Have you dug a hole out there? A foot deep?
I'm just thinking that if your topsoil is only two inches deep, then your lawn is gonna end up kinda crappy anyway - unless you add topsoil.
Do you have access to a tiller?
Can you rough up the soil a bit with a rake?
How cold will it get there this winter? Based on your
answer to this, I could suggest some seed that you can plant that will grow through the winter and smother all weeds in the spring. Then you can cut it down or till it in and then plant your grass seed.
Another option is to till now, plant grass seed, and then your seed and your weeds will be in a race. You will still have some weeds, but good lawn care will eliminate most of them. You could even
skip the till step, but that would mean more weed struggle for longer.