Calculations and predictions are presented within the framework of the statistical hadronization model for transverse momentum spectra of the charmonium states J/$psi$, $psi(2S)$ and $X(3872)$ produced in nucleus-nucleus collisions at LHC energies. The results are confronted with available data and exhibit very good agreement by using particle flow profiles from state-of-the-art hydrodynamic calculations. For $X(3872)$ production in Pb-Pb collisions we predict a transverse momentum distribution similar in shape to that for J/$psi$ with a strong enhancement at low transverse momenta and a production yield of about 1% relative to that for J/$psi$.