  A method for modelling the prompt production of molecular states using the
hadronic rescattering framework of the general-purpose Pythia event generator
is introduced. Production cross sections of possible exotic hadronic molecules
via hadronic rescattering at the LHC are calculated for the $\chi_{c1}(3872)$
resonance, a possible tetraquark state, as well as three possible pentaquark
states, $P_c^+(4312)$, $P_c^+(4440)$, and $P_c^+(4457)$. For the $P_c^+$
states, the expected cross section from $\Lambda_b$ decays is compared to the
hadronic-rescattering production. The $\chi_{c1}(3872)$ cross section is
compared to the fiducial $\chi_{c1}(3872)$ cross-section measurement by LHCb
and found to contribute at a level of O(1%). Finally, the expected yields of
$P_c^+$ production from hadronic rescattering during Run 3 of LHCb are
estimated. The prompt background is found to be significantly larger than the
prompt $P_c^+$ signal from hadronic rescattering.
