I don't see why you need a negative relay. You can still get your switch power from a key on source, run it through the relay coil and then the other side gets ground from the headlight switch. Why do you need a negative relay in the OP's case? What am I missing?
The highbeam signal under the dash is a negative trigger. You woukd use this to power the switch. Ehen the high beams come on the switch is powered. One relay negative trigger to power swtich.
The switch controls the lights ultimately when the switch has power you can press the button and the different lights woukd come on. The switch controls a pos relay to power any light.
So in closing a neg triger relay powers the switch, then a positive relay after the switch to power the lights.
Comes in very handy for the front lights of a vehicle. Your lights, lightbar, bars up front will not work unless you are using the highbeam. Not all do this. Works well if you want to be able to run your lights on backroads and be able to flick them off along with your brights with the pull of the higbeam stalk. Also the ignition has to be on as well to function.
I have my circuit/relay box which I have yet to install under the hood. It is prewired for this circuit with the fuses and relays in a bussman box, my lights are going to be wired as follows. Highbeams/Switch -> Roof Led Lightbar, Highbeams -> Switch Grill Led Lightbar, Ignition -> Switch Side Led Lightbars, Ignition -> Switch Rear Led Lightbars, Igniton -> Switch Rock Lights, 12V Contant -> Fuse -> Strut Switch Under Hood Led, my other electricals in the bussman are as follows for reference Ignition -> Fuse Radar Detector, Ignition -> Fuse Front Recording Video Camera, Constant -> Switch Engel 45, Ignition -> Fuse CB/HAM Radio. Constant -> Switch Spare Battery, in a Bussman 10 Relay Box. You do not have to do this to way to wore up the lights it is just one way.
Alternatively you could do Ignition -> Switch to control Light, or Constand -> Switch ti.
Athough it is truley more like - Highbeam -> Neg Relay -> Fuse -> Switch -> + Relay -> Fuse -> Lightbar. All other circuits follow similar means even the + Ignition/Constant -> Fuse -> Switch -> + Relay -> Light or whatever. I hope you can follow. If not again look at 12volt.com to see how these are wired.
Note: Full disclosure, i did not build my Circuit/Relay Box Skygear on Tacoma Forums did. I looked at other then looked at doing myself cause of how I wanted the items to work and the connections I wanted he had the best solution and built better than I could for same or less than other solutions. It aint cheap but easy to track down if I have any electrical issues.
Excuse me if I got some of it backwards I didnt see but someone will chime in and correct it. I do have a TBI and small detail do escape me as well as my memory is shot I barely remember this stuff. Also, to OP if you use the ground on the highbeam on the relay it is not the correct way of wiring it, will it work yes yet can cause issues.