First, I'm not an electrical engineer--this advice is only as good as the price you paid for it!
Second, please don't use crappy automotive relays--something with a
datasheet and screw terminals would be better, an aircraft part even more so.
Third, about wire size...it never hurts to run the numbers. Assuming 105C rated, tin-plated MIL-W-27759 wire, not in a bundle, below 20,000 feet, ambient temp of 20C, and a 12 foot wire run including ground path...24 AWG wire IS sufficient for intermittent 3A loads. Reference AC 43.13-1B 9/8/98 Ch 11 figs 11-3, 11-4a, and 11-6. That said, I doubt your stick switch can handle a 12VDC 3A inductive load.
Last, I'm a huge fan of HOTAS, but make sure you do some thinking about locking out that switch--we lost a T-38 because someone's pants decided to activate the throttle-mounted speed brake switch.