So how married is your friend to the idea of stereo? Perhaps just run the center/summing channel and one speaker in each of the outboard rooms? I know some amps come with a mono switch.
When I use speakers, I only ever hear stereo when I’m in the sweet spot. When I’m in the next room or other end of the room, mono would be good enough.
The only other thing I can think of is, get rid of the autoformers and use the two leads for the hot part of stereo, implement a truly single ended solution and use the house ground for both channels’ return/ground. Maybe you can use the autoformers to protect the inputs and prevent crosstalk.
One more silly idea — use a broadcast codec, but you’d need two per, one each side.