Electrical – Using 1 momentary switch to toggle between two latched states. Plus relay question

I was really hoping to only connect the circuit to the PC's power switch wires, and not require a separate power supply line.

Is something like that even possible?

Converting a toggle or latching switch, into a momentary output, typically requires some active parts, like logic ICs, a 555/556 timer, a microcontroller or dedicated IC.

Your capacitor + relay circuit is one of the simpler methods, but the pulse will be equal length on both connect/disconnect of the knife switch.

A microcontroller with a simple interrupt code would be easy to setup for your requirements, and only need the microcontroller, a capacitor, and power.

Honestly though, you are already running two wires, one of which is either ground or VCC, a third wire would be trivial.

Assuming that your existing circuit works well enough as a single-output, I'd suggest adding a PNP transistor stage to follow on and add a second inverted output.

^{simulate this circuit – Schematic created using CircuitLab}

For your original circuit, when its light and the LDR has a low resistance, the voltage divider it forms with R2 will turn Q1 on which pulls OUT1 low. When its dark enough and the LDR resistance rises enough to cause Q1 to turn off, OUT1 is then pulled high by R1. The follow-on stage with Q2 acts like a simple inverter using OUT1 as its input. When Q1 is on (and OUT1 is low) Q2 is turned on by the current flowing out of its base through R3. When Q2 is on, OUT2 is pulled high and when Q2 is off OUT2 is pulled low by R4. So although both transistors turn on and off together, the way they're configured causes OUT1 and OUT2 to be in opposite states - much like the SPDT switch you're thinking of.