Assuming that your Taylor expansion/Poisson arguments are enough to handle the case $np$ bounded, I believe the case $np \rightarrow \infty$ can be taken care of using the law of large numbers/Chebyshev. We have $$I_n(p) = E( H(p) - H(X))$$ where $X$ is $\frac{1}{n}Bi(n,p)$. Assume WLOG $p \leq \frac{1}{2}$, and take arbitrary $t<p$. We can bound $H(p)-H(X)$ above by $H(p)-H(t)$ for $t \leq X \leq 1-t$, and above by $1$ in general. This means $$I_n(p) \leq [H(p)-H(t)] + P(X<t) + P(X>1-t)$$
If, say, $t=p-(np)^{-1/3}$ and $np \rightarrow \infty$, then by Chebyshev's inequality the latter two terms are both at most $(np)^{-1/3}$, while the first term goes to $0$ because $H$ is continuous.