Forward checking with conflict directed backjumping
If forward checking detects that the potential assignment of a variable $x_1$ leaves no valid assignments for the variable $x_2$, one simple strategy is to just make the assignment to $x_1$ and then queue $x_2$ as the next variable to be assigned.  When the assignment of $x_2$ inevitably fails, your backjumping machinery will do what it always does and you need not code any special information gathering related to forward checking.  In the meantime, you've saved all the time that would have been wasted doing (and undoing) a potentially exponential number of unrelated assignments that do nothing to resolve the inevitable conflict between $x_1$ and $x_2$.