I'll use the Riemann-Hilbert correspondence between complexes of holonomic $D$-modules with regular singularities and complexes of constructible sheaves.

We can find a stratification such that both complexes are locally constant on each stratum. Since Euler characteristic of hypercohomology is additive in stratification, we may reduce to a single stratum, i.e. reduce to the case when both complexes are locally constant.

For complexes of locally constant sheaves, we can prove this statement using a formula: The Euler characteristic of the hypercohomology of a complex of locally constant sheaves is the Euler characteristic of the stalk times the Euler characteristic of the space. For this statement we can reduce to the case of locally constant sheaves and then calculate cohomology using a cell decomposition, which produces a complex where the dimension of the chains in each degree is proportional to the rank of the sheaf.