it would seem from bill's counterexample that some further constraints on $E$ are needed to get what you want. requiring that $E$ be tight would certainly do the job - but it may be an unnecessarily restrictive assumption.

for example, let $E$ be the [parametric] family of poisson distributions. [so we change the support of the measures to $N := \{0,1,\cdots\}$]. let $\cal P$ denote the set of probability measures on $N$.

for any $Q$, it is pretty clear that as the poisson parameter $\lambda\to\infty$,

$$I_\beta(Poi_\lambda,Q) := \sum_{x\in{N}} Poi_\lambda(x)^\beta Q(x)^{1-\beta}\to 0.$$

so in obtaining $s$, we may restrict attention to some bounded interval $[0,L]$ for $\lambda$.
then, as $I_\beta(Poi_\lambda,Q)$ is continuous in $\lambda$, its max is attained on $[0,L]$.

in this example, $E$ is not tight, altho its only pointwise limit points outside itself are $\delta_0$, the measure putting probability 1 at 0, and the zero-measure $\delta_\infty = (0,0,\cdots)$. [$\delta_0$ should be included if one wants $\lambda > 0$ *a priori*.] here, the only *defective* limit point is $\delta_\infty$.

additionally: altho it is true [as the OP states] that pointwise convergence [the product topology for $\ell_1$] and "total variation" [or $\ell_1$-norm] convergence are equivalent for $\cal P$, the two are not the same for $\ell_1$. in the poisson example, $\delta_\infty$ is a pointwise limit point but not a strong limit point of $E$. [so $E$ *is* strongly closed in this case. i think this was involved in my previous (somewhat hastily conceived) comment.]

in bill's example also, $R_\infty$ is also a pointwise but not a strong limit point of $E= \{R_n: n\ge 2\}$.

these examples suggest that when $s>0$, tightness for $E$ can be weakened to its pointwise closure having at most one defective distribution: $\delta_\infty$. [it seems almost obvious then that if $I_\beta(P_n, Q) \uparrow s>0$ as $n\to\infty$, that any convergent subsequence of $\{P_n\}$ must tend to a limit in $\cal P$. a fancier way to put it is that $I_\beta: P \to I_\beta(P,Q)$ is continuous for the $\ell_\infty$-weak topology on $\cal P$, which is really just pointwise convergence by another name in this case.]

an interesting $E$ satisfying this condition is the set of all binomial distributions on $N$, where both $n$ and $p$ are parameters. here $E$ has lots of pointwise limit points it doesn't contain [like the poisson distributions], but only one defective pointwise limit point: $\delta_\infty$.