  This paper is a sequel to "Representation growth of maximal class groups:
non-exceptional primes". We use a constructive method to calculate some
exceptional cases of $p$-local representation zeta functions of a family of
finitely generated nilpotent groups $M_n$ with maximal nilpotency class. Using
the machinery of the constructive method from the prequel paper we construct
all irreducible representations of degree $p^N$ for all $N\in \mathbb{N}$ for
the group $M_{p+1}$ for a fixed prime $p$. We also construct all irreducible
representations of degree $2^N$ for the group $M_4$. Together with the main
result from the prequel, this gives us a complete understanding of the
irreducible representations of the groups $M_3$ and $M_4$, along with their
global representation zeta functions.
