I haven't used that border crossing, but at immigration and vehicle registration they do ask for your greencard, and if it doesn't cover Turkey they'll send you to the insurance booth anyways, so don't worry about finding it ;-)
I don't know about prices, as my insurance was good for Turkey. If you're feeling lucky, you could ask your insurer for a policy that covers the European part only. At immigration that'll do, and nobody will check where else you roam afterwards (so long as you don't kill anyone).
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