This badass challenge takes place in The Fridge, and is Stiff Competition: Do not kill any Midget Boners before killing Smash Head.
So I've twice now killed Smash Head, leaving his three Midget Boners alive, and not been awarded the badass challenge. I clear the area before opening the loot chest which awakens Smash Head, so I only have to deal with him/them. I then retreat to safety in the entry area and snipe him from a distance. Sometimes he follows me into the entry area, so I have to retreat farther down the hall. Eventually, I whittle his health down and kill him. When he dies, the three Midget Boners are still alive, and come after me, so I know I didn't accidentally kill one of them.
I'm wondering: is the challenge only awarded the first time you encounter Smash Head during the quest to find the Goliath's lost loot crate?
Based on suggestions below, I tried the strategy of staying in the area and performing a close-up kill of Smash Head using a high power shotgun. Same result, the three Midget Boners are alive and still no badass challenge is awarded!. I think I've found a game glitch...