Is there a time that works better for seeding in which you are not likely to get the weeds??? What kind of weed is it???
It kind of looks like a winterweed we have up here... ours will stay tiny and is not a big deal... after the broadleaf app is done and the grass is growing strong it should be done... I never get too excited about broadleaf weeds coming in, because they will always be the weakest plant in a mature lawn, as soon as you spot spray...
You may want to consider adding a sandy compost topdressing after aerating or till in a good sandy compost and start over if this fails... it doesn't make adobe, so don't be sidetracked from building a quality soil base... how deep is your clay layer now???
Now that I know that clay's texture(platelets) has nothing to do with water infiltration, percolation, or drainage,,, I wonder what does...