Your notation is clear to me! It can be shown that:
$D_f(p^{\star },q)=\int dxq\left(x\right)f^{\diamond }\left(f^{\diamond \#}\right(⟨\lambda ,T\left(x\right)⟩\left)\right)$

Even though $f^{\diamond }$ and $f^{\diamond \#}$ are both convex, their composition is not necessarily convex. Sorry, I don't have any clear counterexamples at the ready. (The Hessian determinants all vanish in the graphene/bit-string model.) It's likely that I've configured my numerical solvers badly for this example.

As far as binding to reality:

1. Power laws are ubiquitous in nature. Exponential family doesn't admit power laws; f-div does. (It admits many other important non-exponential distributions as well -- see the B-E example above.)
2. Symmetries in your data that are weaker than -- or just different from -- independence are ubiquitous in nature. Exponential family is only congruent with independence; not so, f-div.
3. KL / Gibbs entropy is inadequate for many observable systems. This typically appears under the heading nonextensive entropy (Gell-Mann & Tsallis w/ contributions by Touchette and others) or Tsallis entropy. This paper is far enough outside my field that I can only loosely validate it, but it's an attempt to explain dark energy by way of applying observational data to determine Bayes factors of different nonextensive entropy models of our cosmological horizon.
4. I consider Jensen's, DPI, and coarse-graining monotonicity to be pretty important desiderata (see Nielsen). f-div is the biggest family that supports these, uniquely determined.
5. [I could write several more posts worth of material]

Zero-math-just-words summary:
So the natural world seems to be full of this symmetry thing, which is great because I suspect otherwise we wouldn't have any hope of making a good map of it. Here's some math that makes constructive epistemic claims about a particularly simple example -- permutation symmetry -- that people have been generally confused about for 90 years. Then here's how that idea would testably interact with a thought experiment that you could almost pull off in a lab (BYO magic box).

Remember T is a program that's supposed to emit all the computable parts of D that are relevant for doing inference on A. That's P(A|D) = P(A|T(D)). (Naturally, the identity function on D does a perfectly fine job of preserving the relevant parts of D, so to be non-trivial you really want T to compress D in some way.) Here $\pi$is some permutation of D (this doesn't compress D). T(D) = T($\pi$(D)) just says that T doesn't care about the ordering of D. Then you combine these facts to resolve that inference on A also doesn't care about the ordering of D. That's P(A|D) = P(A|$\pi$(D)).

Negative example:
$A_{meow}$ = "this sensory input contains a cat"
D = "the pixels of the sensory input"
T = "AlexNet without the final softmax layer"

Observe $\pi$ destroys inference for $A_{meow}$. P($A_{meow}$|D) $\ne$ P($A_{meow}$|$\pi$(D)). Classification of natural images isn't invariant to arbitrary reordering of pixels. AlexNet isn't invariant to reordering of pixels because gradient descent would rapidly exit that region of weight space. T(D) $\ne$ T($\pi$(D)).

Perhaps there are some other (maybe approximate) symmetries ($\sigma$) like reflections, translations, blurring, recoloring, rescaling. P($A_{meow}$|D) $\approx$ P($A_{meow}$|$\sigma$(D)). AlexNet might be in or near those regions of weight space. T(D) $\approx$ T($\sigma$(D)). This should also apply to your cortex or aliens or whatever has a notion of a cat!

I would be very surprised if this didn't connect to natural latents. It connects to universality. I don't think it connects directly to infrabayes but they might be compatible. It also connects to SLT, but for reasons that are beyond the scope of this post.

In the Ising model, the inability -- through local fluctuations, well below $T_c$ -- for the coarse-grained state $⟨\sigma ⟩=-1$ to transition to/from $⟨\sigma ⟩=1$ is an example of superselection. Unfortunately, I can't find a good primer on the subject, but here's something: https://physics.stackexchange.com/a/56570

To be able to take your desired step, reasoning from local->global, you also have to make additional measurements to rule out non-uniform temperature and non-uniform magnetic fields.