A topological group is homogenous. In a finite CW complex, the interior of every top dimensional cell is a topological manifold. Therefore, a finite CW complex which is a toplogical group is a topological manifold. By Hilbert's fifth problem,a topological group which is a topological manifold is a Lie group (not necessarily connected).
Finite CW complexes are also compact, so your group is a compact Lie group.
Positive dimensional Lie groups have torus subgroups, hence have $\chi(G)=0$, so $BG$ is not a finite CW complex as you say.
Zero dimensional Lie groups have $\chi(G) = |G|$, so $BG$ is only a finite CW complex if $|G|=1$.