I defeated Brayko using a combination of methods recommended for other bosses in the game, plus some methods custom to Brayko.
If you have already met Steven Heck (i.e. have done Taiwan missions), and have good reputation with him, then you can buy some help from him. The Clearinghouse Intel choice which supplies Brayko with poisoned cocaine will help you during this boss fight. Brayko won't get "coked up," instead his health will drain during that stage of the fight.
While Brayko is on stage, wait until he is reloading. Use that time to place an incendiary or fragmentation grenade on the wall you are covering behind. Return to cover. Later in the boss fight, Brayko is likely to trigger this trap.
When shooting Brayko on the stage, use weaker ammo in order to save steel rounds or ADPS. Brayko leaves the stage after a set amount of damage. Once Brayko has left the stage, use armor-piercing ammo such as steel core bullets or ADPS.
If you have Shadow Operative for 20 seconds or longer, you can prevent Brayko from finding you for the duration of this fight. Hide in the alcoves on either end of the open area. It is useful to trap the entrances with grenades. An outer incendiary grenade will take down his endurance. A shock-trap within the entrance ensures that Thorton can escape with Shadow Operative or just running behind Brayko's back. If possible, use a Chain Shot during Brayko's paralysis to take down his health, then escape.
I set traps on both alcoves, lured Brayko into each, and Chain Shot him (two shots) with Steel Core bullets, then escaped to the other alcove. This reliably defeated him, using Shadow Operative 20 second duration to escape.
In general, the Chain Shot is very useful against bosses in Alpha Protocol.