I'm looking for a class of groups where there are only two possible conjugacy class sizes, say $1$ and $k$, but at least three distinct dimensions of the irreducible characters. I know the conjugacy class size restriction forces the group to be a nilpotent direct product of a $p-$group and a subset of the center by Theorem A.23.4 from Groups of Prime Power Order Volume 2 by Berkovich and Janko, but I have no idea if this forces only two distinct character dimensions as well.
Ideally (assuming the above is possible) I'm looking for not just a specific example, but a class of examples.
Edit: I ran some GAP code and I've found examples of where this occurs (such as SmallGroup(256, 10070), but it's still unclear to me if there is an easily-described class of groups with this property.
This is not a complete answer. By Theorem A.23.4 you cited, we may assume that $G$ is a $p$-group. By the subsequent results, there exists an abelian normal subgroup $A\unlhd G$ such that $G/A$ is elementary abelian. On the other hand, the groups with character degrees $1$ and $p$ are classified in Theorem 12.11 of Isaacs' book: there exists abelian $A\unlhd G$ of index $p$ or $|G:Z(G)|=p^3$.