The world is still kind of waiting for a 1/4" TRRS jack. If Neutrik made one that was the same size as the current jack we use, it would very likely go in the SEX kit.
The easiest way to do what you want would be to carefully drill out the headphone jack hole to accept a 4 pin XLR for balanced headphone use. This is not that hard to do with a stepped drill bit.
You can then wire up the amp and just connect the wires leaving the output transformers and the wires going to the binding posts to the 4 holes on the XLR jack. It will not switch the speakers off when you plug in headphones.
At the binding post end of the amp, instead of the single jumper resistor between the black post and the chassis, use a 100-250 ohm resistor from each binding post to the chassis. 1/4W is an adequate power rating for this purpose.