I seek an example of an amenable group of finite cohomological dimension that is not virtually polycyclic and has finite abelianization.
Elementary amenable groups of finite cohomological dimension are virtually solvable.
There are lots of virtually abelian groups of finite cohomological dimension that have finite abelianization. The simplest one is the fundamental group of a certain flat 3-manifold, the Hantsche-Wendt manifold.
Virtually polycyclic groups are finitely presented, so perhaps an example can be found among virtually solvable groups that are not finitely presentable, which do exist.
Elementary amenable groups of cohomological dimension two are classified here, see Theorem 3, and they have infinite abelianization (except for the trivial group).
Another idea would be to search for an amenable group that is not elementary amenable. There are very few known examples, and I do not know if any of the examples have finite cohomological dimension and finite abelianization. See here for a list of torsion-free examples (of course, finite cohomological dimension implies torsion-free.