The Thing Cheats

Before you enter the room where you meet Boss 3, push your engineer towards the door until it opens. You will be able to look into the room without the Boss Battle starting. From there you can shoot the tank holding the walker and he will come out to attack you. Get him out and you can kill him. Then when the Boss Battle starts you will have one less Thing to think about.

Before entering the room with Boss 3, push your engineer towards the door until it opens. Then with the door open and your engineer between you and the doorway, take one last step to enter the room and you will push your engineer in too. He will attack the Boss for you and the Boss will attack him instead of you. Until he's dead, that is.

Variants:
Take a couple steps back before pushing the engineer through the door and run at him. Instead of pushing him all the way into the room, you'll go through him and he will stay outside but his face and gun will be visible through the door and he'll still attack the Boss briefly.
Also, if you do this, the door may stay open for the cutscene, during which the engineer will attack through the door. When I did this, he had a shotgun and you could see blood spraying from the back of my head with each shot during the cutscene but it did no damage to me. Not sure if having the engineer attack during the cutscene does damage to the Boss.