How could I build a headphone stereo to mono converter?
I think, before over-complicating stuff with an op-amp, try this: -



It could be that the short circuit detection is using DC to decide whether it is shorted. The dc blocking caps shown are 10uF and in case these do not allow enough bass, be prepared to raise their value to 47uF.

You might also need to have bleed resistors from each input line to shield just to activate the output - try 10k on these.