Spray it. Let it die. Landscape fabric. Add your stone.
In a perfect world: Having a sod cutter like you would see mounted to a tractor at a sod farm. You would be looking at 15 pallets of sod to take away. Which is quite a bit especially considering the slope you described. After you take it all up now you have to haul it all away. What equipment do you have to handle that?
When it's all said and done the steps of tearing out the sod and hauling it away will probably make your customer back out of that idea due to cost.
You could give them two options.
1. Removal of sod. Haul it away.
Spray remaining grass Landscape Fabric. Stone.
2. Spray. Let the sod die. Landscape Fabric. Stone.