This is rather late, and not a full answer to your question, sorry. Nevertheless, some useful information pertaining to the question can be found in a preprint of Annette Huber and Wolfgang Soergel, arXiv version can be found here. The goal of that paper is to compare different integral structures in the top cohomology of $GL_n$: one integral structure comes from the natural $\mathbb{Z}$-form of $\mathfrak{gl}_n$, another from the suspensions of Chern classes in de Rham cohomology, and the third one from the dual of the fundamental class in singular cohomology. They work out the explicit comparison factors. The comparison between de Rham and singular cohomology involves the usual period $2\pi i$, and the comparison between de Rham cohomology and the subspace coming from the Lie algebra cohomology is in fact a rational factor (so no further periods).
As you mentioned periods in your question: there is a philosophical reason why the comparison between de Rham cohomology and singular cohomology should only involve rational multiples of powers of $2\pi i$. If $G$ is a reductive group over $\mathbb{Q}$, we can view it as a variety and as such it has a mixed Tate motive; therefore, all periods should be rational multiples of powers of $2\pi i$.
Similar methods could surely be applied to other Lie groups. The obvious guess would be that the rational factor can be explained by some Weyl group combinatorics in general. I am not that sure if it is straightforward to extend the comparison result to the whole cohomology, but somehow it should be possible to compare the generators of Lie algebra cohomology to the duals of the Chern classes.
Later edit: Another thing that may be noteworthy. The rational factor in the comparison between Lie algebra and de Rham cohomology that is worked out in the Huber-Soergel paper is $\prod_{j=1}^n(j-1)!$. These same factors appear when comparing homotopy and homology of $GL_n(\mathbb{C})$ - the Hurewicz map sends a generator of $\pi_{2j-1}GL_n(\mathbb{C})$ to some multiple of a generator of $H_{2j-1}(GL_n(\mathbb{C}),\mathbb{Z})$, and this multiple is $(j-1)!$. Maybe this is related to the comparison of Lie algebra to Lie group cohomology? Maybe this is the way the subspaces are related - that the generators of Lie algebra cohomology in degree $2j-1$ are a $(j-1)!$-th multiple of the Chern classes?