They have zero incentive to make it cheaper, ever.

Sure they do, competing against other burger joints. There's a reason the prices are set where they are now: they're high enough to generate a profit, but low enough to entice people to come there instead of other burger joints. But, right now those prices are static, not dynamic.

A burger joint that's open late has to pay for staff. If they're just sitting around without any customers they're losing money. If they can lower prices to get people to come to them instead of Mc Donalds or Burger King at night, then they can presumably make more money.