Radiator behind sofa - Box Design
As an alternative, requiring rather less work, how about this:

get some sheets of rigid insulation such as is used in places like ceiling over a furnace (also fireproof but you don't care). Attach these in an unobtrusive way to the back of the sofa.
place some wooden blocks between the radiator (or wall) and the feet of the sofa to act as standoffs.

This way the radiator has plenty of room for convection and the sofa is fully protected.