We investigate the electron-phonon (e-ph) interaction in single-wall carbon nanotube samples at room temperature using femtosecond time-resolved photoemission. By probing electrons from the vicinity of the Fermi level we are able to study the e-ph interaction in the metallic nanotube species only. The observed electron dynamics can be used to calculate e-ph scattering matrix elements for two likely scattering scenarios: forward scattering from twistons and backscattering by longitudinal acoustic phonons. The corresponding matrix elements reveal an intrinsically weak e-ph interaction approximately 50% smaller than predicted by tight-binding calculations.