In my opinion this would much things up by taking a simple topology and making it complex which usually means headaches.
since we are talking straight balanced output to the headphones while retaining the basic circuit even the power supply if beefy enough need not be changed but just used normally to drive two stereo amps instead of just the one.The question then becomes how to get the input of this stereo pair now used as a single balaned channel and there is where you have choices and can decide how far to go with this.
1-the expensive but best soltion would be to use an SE-To-Balanced lne transformer.These babies would set you back at least $50 per channel but offer the ability to use the input either full balanced or single ended while producing a "true" split +/- output to drive the single amp in a balanced mode.
You would want the 15K/15K jensen line transformer if utmost quality is of importor one of the Edcors for a low cost solution (around eight bucks or so).
2-one of the various TI/BB/ADI balanced audio line driver chips.A self contained monolithic solution costing way less than option #1
3-A discrete jfet differential amp.This could be as simple as two N-Channel JFETs (or one N-CH Dual) or as compicated as a full cascoded ccsed differential jfet amp.The borbely Audio xPress JFET articles can be a guide there (pdf download at borbely audio)
I like things simple unless there is a true gain to be had from comlicating things and would personally choose #1 option or #3 option for most such work but in this case I think the monolithic solution is not only fine but maybe the best solution all else considered.
Check the DRV134 chip sheet at TI and have fun,what this is all about right ?