Raman excitation profiles are obtained and compared for carbon nanotube radial breathing mode (RBM) fundamental and overtone vibrations for 5 specific chiralities. Fitting of the Raman excitation data is performed using Raman transform theory. The Huang-Rhys factors obtained from the modeling are directly related to the magnitude of the RBM exciton-phonon coupling element, which is shown to be in a weak coupling limit. The values of exciton-phonon coupling strengths and the possible role of revealed non-Condon effects are in agreement with quantum-chemical modeling.