I'm not clear what you are proposing. But it seems to me that you can leave the secondaries attached (one set to the headphones, the other set to the speakers) and leave the "cold" primary connected to the power tube cathodes, then switch the parafeed cap to one set of transformers or the other. A single DPDT switch would do that, in stereo - or a SPDT for each channel if it's more convenient to install the switches far apart.
Because the parafeed cap is between the high voltage and the transformers, the switch does not need to handle any DC voltage, and if the amp is off there will be no signal voltage either - so pretty much any switch would be fine.
Unless I have misunderstood your question?