The White with black strip is key on positive. It joins the 15A accessory fuse. You can pull power from this for whatever you need while the key is on.
The GRAY wire however, is a switched ground. It goes to the gear position switch. It is ONLY grounded when in reverse.
You method is going to be tricky. I do not recommend grounding the gray wire outside its current circuit. If you do, it could send false single to the computer saying "REVERSE" when in fact it is not. I have not tested to see if it does or not but I think it will.
Oh... Try this, think backwards. Use the ON/OFF/ON in reverse.
Point 1 - IN - gray wire from plug.
Point 2 - OUT - To light negative.
Point 3 - IN - To frame ground.
Light positive wire connects to the white wire with black stripe.
Let me know what you get.