While Carlson points to the old dog's smell and its debilitated condition as cause to put it down, he is also a cruel and rather sadistic man who may well derive some pleasure in killing something.
That he is disgusted with Candy's dog is evinced in Carlson's suggestion to Slim to convince Candy to shoot his old dog, and then give him one of the puppies that Slim's dog Lulu has whelped. To further convince Slim of this idea, Carlson adds, "I can smell that dog a mile off."
Later on, when Candy and the old dog enter the bunkhouse, Carlson, who is already disgruntled over losing at horseshoes, exclaims,
God awmighty, that dog stinks. Get him outa here Candy! I don't know nothing that stinks as bad as that dog. You gotta get him out.
Candy says that he does not notice the smell, but Carlson complains,
I can't stand him in here. . . That stink hangs around even after he's gone.
Because Candy cannot bring himself to shoot the dog he has owned for so long, Carlson offers to shoot himself, crassly explaining to poor Candy how he will execute the dog so that it will not know what has happened.
That Carlson may derive some sadistic pleasure from shooting the old dog is also substantiated by his eagerness for the hunt for Lennie after Lennie inadvertently kills Curley's wife. Carlson wants to shoot Lennie, just as he would shoot a dog, but his Luger is gone. While Carlson may believe Lennie took it, ironically, it is George who has the gun, and he uses it "[R]ight back of the head" on Lennie to prevent him from being locked up somewhere.