Interesting question.@MLG Bradman @Emperor Spriggan what do you think about Bradman?
I think he is not a Zeref's demon as all of Zeref's demons have instinct to kill Zeref (see Natsu) and have etherious form. Bradman never said his Third Seal is his etherious form. He also didn't have instinct to kill Zeref. I guess he is pure demon like demons Mira took over before Tartaros arc (Sitri etc).
Contrary to popular belief, there's really no reason to believe that Bradman is a Demon from the Books of Zeref. I think, like the demons from Galuna Island, he's a natural resident of the Underworld.
The answer lies in what happened after he was defeated.
If he really was a demon from the Book of Zeref, this shouldn't have happened. Reason being, in the Tartaros arc, we saw what happens once the demons from Zeref's Book die. Jackal committed suicide, and his body was immediately sent back to Hell's Core. Tempesta also committed suicide after losing to Laxus, and went back to Hell's Core as well. Ezel was killed by FACE's explosion, and also went back to Hell's Core. That was like a revival point for them. Once Hell's Core was destroyed by Mirajane however, this is what happened to Tempesta after he was killed by Gray.
TL
R: Bradman is a natural resident of the Underworld. Zeref recruited him by deciding to take a walk there. 
Alternatively, if he wanna explain this using mythology terms, when Bradman was fighting Gajeel, he said that he leads souls to the Underworld.
Bradman is like the Ferryman to Hell (Charon). I'm assuming his curse allows him to channel power from the underworld, specifically those bearing the mark of Tartaros. We've seen guild marks channel magic before so it's not unrealistic that they could channel curse power.
Also he is a servant of Zeref, who presumably holds power over all the demons of Tartaros. So it isn't beyond the realm of possibility that the reason he has these powers is tied to Zeref.






