Where to get Key hot source?

[LS short box]

An 04 LS should be a 24 tooth crank wheel so I think you got it right.
But a stock C10 brake light switch only has one set of prongs? and when you step on the brake the circuit is closed and the brake lights go on.
Using the 02 Camaro brake light switch (some use 88 Monte Carlo) has two sets of prongs. One set the closes the circuit and one set that opens the circuit. You can check to see which is which with a meter.
It's OK to use a relay but I never have and never had a failure.