Assuming local anesthetic was used to minimize the pain, a cut to the femoral artery might do it quickly. Google search says it only takes 3-5 minutes, sometimes less depending on the severity of the cut. Might be able to hasten things if you cut the femoral in each leg, too
Exsanguination itself? No - it starves your brain of oxygen so you fall unconscious pretty quickly with enough flow.
Getting enough flow to lose consciousness quicky? That’s painful.
Assuming local anesthetic was used to minimize the pain, a cut to the femoral artery might do it quickly. Google search says it only takes 3-5 minutes, sometimes less depending on the severity of the cut. Might be able to hasten things if you cut the femoral in each leg, too