Whatever you decide, please do not use methyl bromide to fumigate the greens. There is strong evidence that this gas does harm the ozone layer. It is being phased out, but not fast enough. Certain farm crops, like strawberries, depend on it, but I would not trade the earth's atmosphere for pure greens.
I had good results sterilizing soil with a granular product, Basamid, that is also much easier to use than methyl bromide.
This was to eradicate bermuda grass, one of the most persistant weeds.
In a cool-season climate, the lasting effects of any sterilization are minimal, I believe. In a matter of months, the greens will be contaminated again with all the pests you've just eliminated. Maybe you can keep Poa out, but it won't be because you fumigated, but rather from the post-seeding management.