Bram's group appear to be a third pillar in the conflict for the Island, working directly for Jacob in a fashion not unlike special forces. It is suggested by Jacob's visit to Ilana in hospital, that this group has been hand-picked by Jacob himself. Though Bram indeed implies that Widmore's team are the "Bad guys", Widmore does seem to be working against The Man in Black and ergo for Jacob too, but it is perhaps his more selfish intentions and manner of execution that come across as antagonistic to Bram's team. The true opposition are The Others, who, originally working for Jacob, have been unknowingly manipulated by The Man in Black for many years. This does not become clear to Bram's team until the real John Locke's corpse is revealed.
Ben would want to kill Caesar so that his people could take control after he left for the main Island. Also, Ben would be the 'opposing' side off-Island back when Widmore was doing his recruiting - this is how he got Michael onto the boat.
In all references to the Temple, it is shown to have a strong connection with the Others. "Our Temple" is said quite a few times in the recent past. If this is true, then they would have to be more of a duo, branching from an original group of three (Ilana, Bram, Richard) disciples, so to speak, of Jacob.
More than one group could think the Temple belongs to them. Bram's group as well as the Other's believe that the temple is theirs. It could be a similar situation to the Jews, and Muslims all considering Jerusalem their Holy city.
In Ilana's flashback we see Jacob asking for her help. Later in one of Hurley's flashbacks we see Jacob expressing his wish that Hurley would also go back to the Island. Both of these flashbacks reinforce the idea that they most likely are not working for Ben and are in fact working for Jacob. So Ilana most likely arrested Sayid because Jacob wanted him to come back to the Island.
Bram's Knowledge of the Island
It does seem like he and Ilana know something others don't. The way they ask "What lies in the shadow of the statue" makes it seem like they're not sure what the answer will be. But once they hear that answer they immediately know something about the person. It's my opinion that they're more confident about their purpose than Ben or Widmore, who were vague and manipulative. They offer only knowledge and fulfillment, but you're either with them or not.