  Nonleptonic $B_c$ decays to $P$-wave charmonia and light $\pi^+$, $K^+$, and
$\rho^+$, $K^{*+}$ mesons are analysed using factorization. The hadronic form
factors parametrizing the $B_c \to \chi_{cJ}(h_c)$ matrix elements are
expressed in terms of universal functions at the leading order of an expansion
in the relative velocity of the heavy quarks in the $B_c$ rest-frame and in
$1/m_Q$. Several ratios of branching fractions are evaluated, and when
experimental information can be used, single branching fractions are presented.
Both the $1P$ and $2P$ charmonia are considered. If the exotic candidate state
$\chi_{c1}(3872)$ corresponds to $\chi_{c1}(2P)$, it should be produced in
nonleptonic $B_c$ decays with predicted abundances with respect to the other
states in the charmonium $2P$ spin four-plet.
