I am about to build a phase flip switch into one of the output of the splitter kit from musikding. I am using isolated jacks on that build because tha jacks that came with the kit didn’t even hold the ts-plus in place. The tip-jacks were to short. But my thought is to isolate the switch from the enclosure to ensure that the 2 output never connects. Is your an aby box? Could it be a grounding issue?
Did you proceed with this yet? As stated above, the Musikding wiring will give you that loose cable hum on the output that's not in use since it never shorts anything to ground. You will be much better off using the 3PDT solution that I posted.
I have now rewired mine using said method and it doesn't exhibit any hum. However it still gives the same issue when using the Y-splitting in conjunction with the phase shift (the non-flipped output drops in volume). I'll try and do a more proper explanation of my circuit.
First off the signal hits the first opamp stage of this circuit (the buffer). Pin 1 and R3 is not connected.

The signal from pin 1 and 2 goes through a 220n decoupling cap to the input of this ABY-wiring:

The output going to jack B is routed via a DPDT switch which sends the signal either straight to the jack or to the jack via the phase flip stage (into R3/out via another 220n output cap coming from pin 7/R4).
This works fine until you use the Y-functionality along with the phase flipping on output B, which results in a drastic volume drop on output A. In A/B mode it works just fine with the phase flip engaged, and without the phase flip the Y mode works fine.
I still think that there's no issue on how either circuits operate but rather a problem with the combination of the Y mode and the phase flip on output B. I just cant understand why there shouln't work. Nothing seems wrong with my board and wiring and I've cross referenced it with the original schematic multiple times without finding anything that doesn't match. Do you still want voltages and pictures?