Electrical – Wire laptop power switch to another switch

You were measuring the DC resistance of the various loads in your house in series with the secondary of the transformer your house is connected to, via the "neutral" connection, which is not opened by the breaker. It isn't surprising that this amounted to a few ohms. If you really want to make that measurement across the main breaker, you also need to open all of the branch circuit breakers in the panel as well.

BUT (and this is important!), you should not be relying on the main breaker for this kind of isolation. You must use a proper transfer switch, which will completely eliminate the possibility of ever connecting the generator to the external mains.

There are two general types of transfer switches you can buy. The first is a "whole house" transfer switch, which gets installed "upstream" of the main breaker and switches the entire panel between the mains and the backup generator. The other type is essentially another distribution panel dedicated to the generator, and has individual switches for the branch circuits you want to be able to connect to the generator. I use the latter type in my own house.

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.