Van der Waals equation of state plot limitations

The Van der Waals equation is a model. It's popular because it gives physically realistic predictions without too much work, but it's rarely used in serious modelling because it doesn't describe real substances with especially good accuracy. Overall: it is not surprising that the model doesn't model propane correctly as there really isn't any substance that the model describes especially well at all conditions.

The $b$ term in the Van der Waals equation represents the volume of the particles (which is treated as zero in the ideal gas law). As you observe, $Prightarrow infty$ as $1/rhorightarrow b$; this represents the fact that, once there is no space between the particles, no amount of force can shrink the system further (the model assumes that the gas is composed of infinitely hard particles). If propane followed the Van der Waals equation exactly we would find that $b = frac{1}{3rho_c}$ i.e. the density of the particles themselves would be $3$ times the critical density.

As $rho rightarrow 0$, the behaviour of real gases, Van der Waals gases, and ideal gases all converge. All reasonable models should agree in this limit. As $rho rightarrow infty$, non-ideal behaviours become more significant and the three models begin to diverge. For the isotherm that you plotted, the difference between the real pressure and the prediction of the Van der Waals model begins to grow very quickly around $rho = rho_c$, but to my knowledge this is just a coincidence. The exact point at which the model breaks down is likely different for different isotherms.