Common solutions to 3SAT and 2SAT models comprised of the same variables
No, in general you cannot.  If $S_2(L) = (x_1 \lor \neg x_1)$, say, then the implication graph for $S_2(L)$ gives you no information on the solutions to $S_3(L)$ or $B$.
If you want to know how to solve it in practice, throw $B$ into a SAT solver.  No need to distinguish between the two types of clauses: SAT solvers already do everything there is to do.