1997 GMC K2500 Suburban - no brakes in tow harness

Actually where your stop lights turns into blinker lights is in the "signal light switch". When a lamp goes to blink it is disconnected from the brake circuit for that side of the vehicle. You should not have a problem with overloading a wire, as draw is very low. Also before they had a special sets of wires for a trailer, you always wired into the circuit at the tail light.