If you dont have a stair chair... what about a reeves? a backboard? take one flight at a time and rest for a minute before descending the next. The FD was there... have them carry the pt. And if the pt. refuses to be carried... document it and make sure you also document that the associated risks of that action were explained to the pt. Even if a pt. refuses a specific treatment, its an obligation of the EMS provider to specifically inform the pt. of all associated risks so the pt. can make an informed decision. Because a refusal by a pt. can be disregarded completely if they were not informed of the risks involved.
Now that would be a tough call. A reeves, backboard or breakaway transfer flat would be a tough choice over walking. In general they are extremely unstable since the center of gravity is above the support. Additionally, unlike a stair chair and basket stretcher, there's nothing besides the straps preventing the patient from sliding off. A basket stretcher would definitely work better, but I sincerely wonder if the stress of being strapped and carried in something so unstable would be worse than the exertion of walking.