Too many nuts, too little lawn

There is a stand of mixed hardwoods, including mostly mature and extremely prolific (hickory?) nut trees, beginning about 20 feet south from my back porch. I would like to have a strip of lawn between my porch and the trees, but the yearly onslaught of nuts combined with lots of shade from the trees has made it impossible. I am in AL zone 7b and have burmuda lawn out front.
Is there any type of lawn that can take this abuse or should I resign myself to just paving in as much patio as I can afford? A garden supply suggested sodding zoysia, but they had pallets of the stuff to sell ;-)

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