Using a window air conditioner outside and redirecting the cold air into room with ducts
I saw this set up you are thinking about doing in 1968 at a donut shop in Pittsburgh, Pa. and it worked okay. (wow how time flies) They did a lot of stupid things then to save a few dollars. The duct work was oversized to reduce friction loss for both the supply air and the return air. Servicing this unit was extremely hard and very expensive since the duct had to be disassembled every service call. After a few years and many very expensive service calls they made the correct decision and installed a different system. Sometimes cheap ideas are just cheap and dumb ideas.  "My 2 cents".