Non-Gaussianness is an ambiguous concept. In the continuum of
probability distributions such as the uniform, where all events are
clustered into a given range and equally likely. On the other side are
structured, spiky distributions with the certain event being the
extreme example.[1]

Therefore the measure of spikiness is usually based on estimators of a shape parameter of an assumed data generating distribution. For example, If you assume the data is generated from a beta distribution, then the spikiness can be measured by an estimator of its shape parameter. This is the classic thinking when a parameterized model is assumed for the underlying probability distribution that generates the model.

Following this idea, a classic test of comparing how similar two probability distributions are is the Kolmogorov-Smirnov test. It induces a nonparametric measure of similarity, and therefore could be used for exploring spikiness. In this direction of characterizing spikiness. In other words, spikiness can be measured by an appropriate choice of norm on the space of probability distributions supported on $[0,1]$.

To be honest I think this is more like a reverse Schwarz inequality rather than a Jesn inequality since I do not see how convexity comes into play. If that is the case, then such a sufficient condition reduces to a choice of $S$ such that majorant conditions hold. For any isotonic functional $A$, including most norms, $0\leq A(f^{2})A(g^{2})-A^{2}(fg)\leq\frac{1}{4}(M-m)^{2}A^{2}(g^{2})$ where $m\cdot g\leq f\leq M\cdot g$
In this case we can take $f=g$ and see if we can related the majorant coefficients $M,m$ with the $\lambda(S)$, which I believe is a common pratice in deriving a bound since the above inequality provides a sharp bound.

[1]Gray, William Charles. Variable norm deconvolution. No. 19. Ph. D. thesis: Stanford University, 1979.

[2]Dragomir, Sever S. "Reverses of Schwarz inequality in inner product spaces with applications." Mathematische Nachrichten 288.7 (2015): 730-742.