Robert, your diagram was very useful.
Your light switch has three wires because it has a built in warning light that requires a separate ground connection
(the black wire). The two switch contacts are connected to the red and white wires, so splicing them together would simply complete the lighting circuit without a switch, resulting in your lights being on continually.
If you carefully study the following diagram, you will hopefully be able to understand how your existing switch is currently wired, and what needs to be altered to connect the GM throttle switch in place of your existing one:
It might be easier to leave the existing light switch in the circuit, and simply join the brown and yellow wires to the red and white wires on the back of the switch. By doing this, your old switch would still light up when the lights were switched on from either switch, but you would also need to make sure that the switches were both in the off position before the lights would turn off.
If your lighting circuit is not fused, I recommend that you fit an in-line fuse to protect the wiring and switches from damage in the event of a short circuit anywhere on the light wiring.
I hope the diagram has made things a bit clearer for you.
Alan