  The family of Shallit sequences consists of the Lucas sequences satisfying
the recurrence $U_{n+2}(k)=(4k+2)U_{n+1}(k) -U_n(k),$ with initial values
$U_0(k)=0$ and $U_1(k)=1$ and with $k\ge 1$ arbitrary. For every fixed $k$ the
integers $\{U_n(k)\}_{n\ge 0}$ are distinct, and hence for every $n\ge 1$ there
exists a smallest integer $D_k(n)$, called discriminator, such that
$U_0(k),U_1(k),\ldots,U_{n-1}(k)$ are pairwise incongruent modulo $D_k(n).$ In
part I it was proved that there exists a constant $n_k$ such that $D_{k}(n)$
has a simple characterization for every $n\ge n_k$. Here, we study the values
not following this characterization and provide an upper bound for $n_k$ using
Matveev's theorem and the Koksma-Erdos-Tur\'an inequality. We completely
determine the discriminator $D_{k}(n)$ for every $n\ge 1$ and a set of integers
$k$ of natural density $68/75$. We also correct an omission in the statement of
Theorem 3 in part I.
