I think if Burger keeps playing like he is, defensively AND with the bat, the Sox should keep him. He's playing great defense at 3B. Try him at 2B and keep your guys. I know it's really early and it's a risk, and there isn't much time left to make trade decisions, but they've made it this far without outside additions. Eloy and Robert are coming back soon too, which helps a lot.
And if Burger doesn't work at 2B (different set of athletic skills needed for that position), I know many people would hate this, but it used to be Moncada's primary position. I know Moncada is better when he needs to get rid of the ball quickly and make quick movements on plays, but maybe he's grown to the point where he'd play 2B well. Then you could move Burger to 3B where we have seen that he can hold his own and more over at the hot corner.