We investigate the link between the bar rotation rate and dark matter content in barred galaxies by concentrating on the cases of the lenticular galaxies NGC 4264 and NGC 4277. These two gas-poor galaxies have similar morphologies, sizes, and luminosities. But, NGC 4264 hosts a fast bar, which extends to nearly the corotation, while the bar embedded in NGC 4277 is slow and falls short of corotation. We derive the fraction of dark matter fDM, bar within the bar region from Jeans axisymmetric dynamical models by matching the stellar kinematics obtained with the MUSE integral-field spectrograph and using SDSS images to recover the stellar mass distribution. We build mass-follows-light models as well as mass models with a spherical halo of dark matter, which is not tied to the stars. We find that the inner regions of NGC 4277 host a larger fraction of dark matter ($f_{\rm DM, bar}\, =\, 0.53\pm 0.02$) with respect to NGC 4264 ($f_{\rm DM, bar}\, =\, 0.33\pm 0.04$) in agreement with the predictions of theoretical works and the findings of numerical simulations, which have found that fast bars live in baryon-dominated discs, whereas slow bars experienced a strong drag from the dynamical friction due to a dense DM halo. This is the first time that the bar rotation rate is coupled to fDM, bar derived from dynamical modelling.