I believe the sts kits activate the buzzer when there is too much oil pressure, backed up, not voltage.
So if you're using the sts stuff I'm not sure how that's gonna work out, to power the LED when the pump stops.
When that small pressure sensor on the pump sees more than 1psi of oil pressure, it trips and the buzzer sets off, or it can trip your LED.
You can have power to the pump but no flow, that's why they use pressure activated buzzer. I think you need to go the same route as sts' idea.