Let the product $(12)(15)(16)$, each factor written in base $b$, equals $3146$ in base $b$.  Let $s=12+15+16$, each term expressed in base $b$.  Then $s$, in base $b$, is 
$\textbf{(A)}\ 43\qquad \textbf{(B)}\ 44\qquad \textbf{(C)}\ 45\qquad \textbf{(D)}\ 46\qquad \textbf{(E)}\ 47$
Converting everything into base $10$, we have $(b + 2)(b+5)(b+6) = 3b^3 + b^2 + 4b + 6$.  Looking ahead, the constant term of the polynomial will be $2*5*6 -6 = 54$. By the Rational Root Theorem, the only possible integer roots are $1, 2, 3, 6, 9, 18, 27, 54$.  Bases $1, 2, 3, 6$ do not have a $6$ as a digit.  Testing $b=9$ gives a solution that works.
Therefore, we are working in base $9$.  Adding the units place in base $9$, $2 + 5 + 6 = 14_9$, so we carry the $1$ to get a total of $44_9$, which is option $\fbox{B}$.