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Bailleul continues his sentence: "... so we provide a proof in an appendix" (p.4). If you look in the appendix, you find the independence assumption right at the beginning (p.29). Also in Theorem 1.2.
I have expanded, for example by adding the definition of a lattice. Unfortunately, the English wikipedia treats only lattices of full rank $r$, The German wikipedia <de.wikipedia.org/wiki/Gitter_(Mathematik)> is more accurate. I looks like some motives are shared between your proof and mine, but we are using different languages. What I call a linear equation (with integer coefficients) seems to correspond to what you call a character.
strange that @A.Bailleul writes "It seems a reference to a proof of the discrete version of the Kronecker-Weyl theorem is hard to find in a published form". totally unaware of the classical Kuipers-Niederreiter book, or the more recent monograph by Drmota and Tichy from 1997, or by Hlawka 1984. (Or why not the proof by Weyl himself?)
Here is a proof that $G$ is not contained in the symmetry group of a duoprism. We look at great circles that are kept elementwise fixed by nontrivial elements of $G$: A symmetry that rotates a plate $P$ leaves the whole circle $F_i$ on which $P$ lies pointwise fixed. Hence there are 12 great circles that are left fixed by elements of $G$. On the other hand, there are only two choices of a circle that is pointwise fixed by a nontrivial symmetry of a "large enough'' duoprism (the two "axis circles'').
