We make predictions for the expected multiplicities of exotic charmed hadrons and charmed nuclei in Au+Au collisions at SIS100 and Pb+Pb collisions at LHC beam energies, using input on light hadron and charm production from the UrQMD transport model, and applying the Thermal-FIST model. We demonstrate that the CBM experiment has the capability to explore these states with production rates of 1 per second for χc0(1P) and χc1(1P), and 3 per minute for X(3872) at the expected data taking rates. These rates should be seen as order-of-magnitude estimates due to the extrapolation of the charm production mechanisms into the unknown high-baryon density and subthreshold territory that FAIR explores. Due to the higher baryon density at CBM compared to the LHC, charmed nuclei, if they exist, will be equally abundant at CBM as at the LHC, even though the total charm production at CBM is much lower.