Yeah I agree with Mick. Get a bobcat with a rake instead of a bucket to go over the whole area. Before the weed seeds would be great. The bob cat rake would bury half of that weed too deep for it to be an issue again and what is left on the top you could rake off.
Depending on how much you could get tree mulch for, I would cover it with a thick layer (like 100mm/200mm) then just spot spray the weeds coming through...
Limiting the light for the germinating seed with a thick wood mulch works well.