Needed a specific relay for add-on reverse lights

[LateOutDrsMn]

So I'm adding some additional lights to the rear of the aveo. The factory lights just don't light up the driveway enough when backing in. Worked well enough when we lived in the city, but since moving a out to the country, it just doesn't cut it. So I found some cheap LED lights. Found a good mounting position. I wired them using a normally closed 5 pin SPDT relay. They take acc switched power from the fusebox, not stealing from the already weak factory lights. I set them up to come on when the factory reverse lights come on. I also added a 3 way switch which allows us to manually turn them on or completely off, regardless of whether the car is in reverse or not. So the switch and the lights work great when using the manual on/off. But the auto on with the factory lights hasn't worked, and I figured out why. I tapped the factory reverse wire near the passenger front door, just before it turns toward the shifter. If I use the multimeter, I can see the change in voltage when shifting in and out of reverse. So I tapped this wire to act as the trigger for the relay. I ran the wire through a one way diode to prevent back feeding the factory lights. The problem is the voltage to the factory lights is only around 7.5, and after going through the diode and all the way to the relay behind the dash, the voltage drops to just over 5 volts. This isn't enough voltage to trigger the coil in the relay. From what I've been able to research online, most of.these relays trigger at just over 7 volts or 60% of the circuit they were set up for; in this case 12v.
So I need a 4 pin SPDT normally closed relay, that works on a 12v circuit, but has a coil that triggers on just under 5v. Anybody know where to.find one?
Thank you

[Eg.h2o]

How about a 6v relay? Or look into triggering it at the gear selector?

[LateOutDrsMn]

So the 6v relay most likely won't last, because it'll be pulling 12v from the fused power.
And originally I was looking to use the shifter to activate the trigger. I was in the shifter replacing the parts for the ignition lock anyways. I couldn't find anything that made the signal unique to reverse. That's why I decided to trace the wires from the reverse lights forward.
Guess I'm still looking for a solution. Thank you

[EAD15]

Have you checked Mouser? They have almost every type of electronic component you could want. Here's a link to their "relay" options

https://www.mouser.com/c/electromech...ors-solenoids/