If we are looking solely at costs, a compromise would have to involve some degree of reduction in the costs associated with the death penalty. The only way to achieve a reduction in costs would be to limit the number of appeals that any one prisoner could file. The high price of the death penalty comes about largely because of the number of appeals that can be filed. Every appeal generally means that the government has to pay lawyers who will argue for the death penalty and those who will argue for the (indigent) prisoner.
This would be a very difficult compromise to make since a reduction in appeals could make it harder for defendants to get justice. However, this is the only feasible way in which to reduce the costs of imposing the death penalty.