Brake lights do come on.Now I know this is going to sound stupid but have you check to see if your brake light come on when you step on the brakes? If they do not this will cause a no crank condition on the vehicle. I believe the key has to be turned to the ON position for the brake lights to come on.
Tried this did not seem to change how long it runs after starting by jumpering the starter.The other thing is when you are bypassing the startup sequence by jumping the starter solenoid the TDM has NOT been instructed to send the fuel continue password so it is not going to do that. You cannot start the car and have it stay running by jumping the solenoid unless the key is turned to start at the exact same time that you jump the starter solenoid. This way the TDM is going to know to transmit the fuel continue password to the BCM. If you have 2 people you can try doing that and see if the vehicle stays running. It should if you get the timing correct. TDM = Theft Deterrent Module which is the passkey 3 and passlock 2 controller.