I don't think they "have" to pay the higher interest. Wouldn't they ultimately get that interest from parking money at the Fed over night? A bank isn't going to pay interest on money that results in a net loss, either they get it from the Fed or they are making private loans at a higher rate. Otherwise there would be no incentive to increase deposits as each would result in lost profits.
A bank either pays competitive deposit interest rates, or people move their deposits elsewhere. That can happen quite quickly, and they'd have the same liquidity problem.
