Where does the backflush line come out? Can you push something in to the back end like a roto rooter type thing and maybe break lose whatever is blocking?
Once a long time ago, my DH, the plumber, was flat on his back on the floor in the living room, not able to get up because he had thrown his back out. Of course it was then the kitchen drain decided to clog up. I turned on the basement water real hot, hooked up a hose and put one of those cone shaped sprayers on the end. I opened the cleanout in the pipe, turned on the water, crimped the hose real tight, adjusted the sprayer to a strong straight stream, shoved the sprayer down the pipe while holding a 5 gallon bucket beneath the cleanout in case the water backed up. I shoved the sprayer until I felt it wouldn't slide any longer and uncrimped the hose. I kept pushing on the hose and sliding it back and forth a bit. IT WORKED!!! That drain hasn't clogged up since either. LOL