This might work. But it also might not depending on the toggle switch and the debouncing behind the pins on the Pi. It also won't work if the Pi is looking for the momentary pushbutton being released rather than being pressed.
It relies on the time between the toggle switch breaking the original connection and making the new one to imitate a momentary push button. It relies on the Pi remembering whether it is actually supposed to be off or on.
This means that if the toggle switch is in the wrong state when power is lost and then restored (maybe replacing batteries), then the on/off position of the toggle switch will be reversed.
It doesn't make a true pulse but I don't think you can make a true pulse without consuming power on some level all the time. What it does is more like if you pressed the button to turn it on and never let go, only releasing it to push it to turn it off and then never letting go again except to push it to turn it back on.
simulate this circuit – Schematic created using CircuitLab
EDIT: I just assumed you had a SPDT toggle switch, but you have a SPST switch. From the voltage measurements you have given between the pins and relative to ground, the following should be what is on the other side of the pins. But I fiddled with it a bit and can't make it work with a SPST. There's just not enough degrees of freedom. A large cap in series with the switch will make it pulse low momentarily when the switch is closed, but I can't think of a way to make it to the same when the switch is opened.
simulate this circuit