You should be able to do something similar with a single stereo jack.

Wire the neck pickup vol pot output to the ring terminal on the jack with a dpdt switch in line in place of the third pot, and the normal output path via pickup selector switch to the tip. Wire the switch so that one side makes a connection between the neck pickup vol pot output and the ring terminal and the other side makes a connection between the neck pickup vol pot output and the pickup selector switch.

Stereo cable with switch set gets you bridge pickup to one amp (assuming pickup selector allows it) and neck pickup to the other.

Mono cable and switch unset gets you normal output path.

Saves running more cables to trip over. If you want to do things like neck + bridge to one amp and neck to the other then it would get a bit trickier.