I too was put off by the appeal to quantum mechanics.

Here is my humble contribution. Let $\Omega(n,V,E)=\Omega_q(n,V)\Omega_p(n,E)$. I assume the factor $1/n!$ stems from $\Omega_q$, the configurational source of entropy.

The number of ways of putting $n$ identical objects into $k$ distinguishable boxes is
$$ N_{k,n}={n+k-1\choose k-1}=\frac{\Gamma(k+n)}{n!\Gamma(k)}.$$ We can make $k$ really big, like $k>n^2/(2\varepsilon)$ and why not? We have point particles and space is infinitely divisible - continuous. If we make $k$ this large, then the fraction of configurations with boxes containing more than one particle is at most $\varepsilon$. 
Anyway, as $k\to\infty$
$$ \frac{\Gamma(n+k)}{n!\Gamma(k)}\sim \frac{k^n}{n!}$$ 
(Note the similarities with the birthday paradox). So increasingly (with ever larger $k$) we have boxes with either zero or one objects. So, for some finite volume $V$ and $V_0=\lambda V/k$, we could say $$ \Omega_q(n,V)=\frac{\lambda^n}{n!}\left(\frac{V}{V_0}\right)^n.$$