Multivariate Bessel processes, otherwise known as radial Dunkl processes, are stochastic processes defined in a Weyl chamber that are repelled from the latter's boundary by a singular drift with a strength given by the multiplicity function $k$. It is a well-known fact that when $k$ is sufficiently small, these processes hit the Weyl chamber's boundary almost surely, and it was recently shown by the authors that the collision times for the process of type $A$, also known as the Dyson model (a one-dimensional multiple-particle stochastic system), have a Hausdorff dimension that depends on $k$. In this paper, we use the square of the alternating polynomial, which corresponds to the reflection group of the process, to extend this result to all multivariate Bessel processes of rational type, and we show that the Hausdorff dimension of collision times is a piecewise-linear function of the minimum of $k$, but is independent of the dimension of the space where the process lives. This implies that the Hausdorff dimension is independent of the particle number for processes with a particle system representation.